University of Ljubljana
Faculty of mathematics and physics
Department of Physics
Gregor Vidmar
The influence of Tunneling on
Eigenvalue Statistics and Dynamics
in the Mixed-Type Systems
Doctoral thesis
ADVISER: Prof. Dr. Marko Robnik
Ljubljana, 2008
Univerza v Ljubljani
Fakulteta za matematiko in fiziko
Oddelek za fiziko
Gregor Vidmar
Vpliv tuneliranja na statistiko in dinamiko energijskih stanj sistema
meˇsanega tipa
Doktorska disertacija
MENTOR: prof. dr. Marko Robnik
Ljubljana, 2008
CAMTE
This PhD thesis has been performed at the
CAMTP - Center for Applied Mathematics and Theoretical Physics,
University of Maribor
CAMTE
To doktorsko delo je bilo izdelano na
CAMTP - Centru za uporabno matematiko in teoretiˇcno fiziko
Univerze v Mariboru
i
Abstract
In this thesis we have studied dynamical tunneling and random matrix theory in the field of quantum chaos.
First, we have considered the spectra of quantum Hamilton systems characterised by the mixed-type classical dynamics. In the semiclassical limit the Berry-Robnik (BR) statistics applies whereas at larger values of heS one can notice deviations from BR due to localisation and tunneling effects. We have derived a 2-level random matrix model which can be treated analytically and the iV-level random matrix model which has been treated numerically. Both models describe tunneling effects. The coupling between the regular and the chaotic levels due to tunneling is assumed to be Gaussian distributed. The results are predicted to apply in mixed-type systems at low energies. The two-level model describes many features of large matrices. The proposed 2 and the N level spacing distributions have two parameters, the BR parameter p, characterising the classical phase space, and the coupling parameter a. The same procedure has been followed for the all-to-all couplings as well. Second, we have studied the mushroom billiard introduced by Bunimovich, especially in terms of its level spacing distribution, avoided crossing distribution and dynamical tunneling rates. The mushroom billiard has a nice property of sharply divided phase space into precisely one regular and one chaotic region. By changing the appropriate parameter one can easily identify the regular or the chaotic states without considering the Wigner functions. First we have studied the level spacing distribution to test the random matrix model and, then, we have focused on the avoided crossings appearing between the regular and the chaotic states. The splittings, appearing at such avoided crossings, indicate the strength of dynamical tunneling. Larger splitting presuppose stronger tunneling effect. With the help of Fermi’s golden rule we have calculated the dynamical tunneling rates for each regular state of the mushroom billiard. We have compared the results from the microwave experiment and those from the expanded boundary integral numerical method with the already-existing analytic prediction derived by Ketzmerick and his co-workers. We have observed excellent agreement in the accessible regime of low energies. Third, we have discussed various ensembles of real symmetric matrices with the dimensions N = 2 to N = oo for a variety of distributions of matrix elements. For N = 2 there exist the exact analytic results obtained by Grossmann and Robnik whereas for N = oo one can rely on the theoretical findings by Hackenbroich and Weidenmu¨ller (HW). According to these findings the local spectral fluctuations are exactly described by the GOE if the limiting distribution of the eigenvalues is smooth and restricted to a finite interval. We have numerically shown, according to HW, that such a transition to the universal behaviour is pretty fast and that it does really not occur if one or both of the conditions from the theory mentioned are not fulfilled. We have tested this findings for the box, the exponential, the Cauchy-Lorentz and the singular (power law times exponential) distribution function of matrix elements.
ii
PACS numbers:
02.50.Cw Probability theory
02.70.Pt Boundary-integral methods
03.65.Sq Semiclassical theories and applications in quantum mechanics
03.65.Xp Tunneling, traversal time, quantum Zeno dynamics
05.45.-a Nonlinear dynamics and nonlinear dynamical systems
05.45.Mt Quantum chaos; semiclassical methods
05.45.Pq Numerical simulations of chaotic systems
Keywords: dynamical tunneling, regular, chaotic, mixed-type system, energy levels, level spacing distribution, level dynamics, mushroom billiard, avoided crossings, tunneling rates, random matrix theory, microwave experiments
iii
Povzetek
V tej disertaciji študiramo dinamično tuneliranje in teorijo naključnih matrik na področju kvantnega kaosa.
Najprej obravnavamo energijske spektre kvantnega Hamiltonskega sistema z mešano klasično dinamiko. V semiklasični limiti, ko je efektivna Planckova konstanta heS dovolj majhna, velja statistika Berryja in Robnika (BR), medtem ko pri večjih vrednostih freff (manjših energijah) vidimo odstopanja od BR zaradi lokalizacije in tunelskih efektov. Izpeljemo 2-nivojski model naključnih matrik, ki ga lahko obravnavamo analitično in iV-nivojski model naključnih matrik, ki ga obravnavamo numerično. Oba modela opisujeta tunelske efekte. Predpostavljamo Gaussovo, eksponentno ali škatlasto porazdelitev tunelske sklopitve med regularnimi in kaotičnimi nivoji. Rezultati se predvidoma uporabljajo v mešanih sistemim pri nizkih energijah. Dvonivojski model opisuje večino lastnosti večjih matrik. Predlagani 2- in iV-nivojski porazdelitvi po razmikih med sosednjimi nivoji imata dva parametra, BR parameter p, ki opisuje klasični fazni prostor, in sklopitveni parameter a. Isti postopek naredimo tudi v primeru sklopitve vseh nivojev z vsemi.
Nadalje eksperimentalno in numerično študiramo gobasti biljard, ki ga je vpeljal Bunimovich, posebej porazdelitev po razmikih med sosednjimi nivoji, porazdelitev izognjenih križanj in koeficiente dinamičnega tuneliranja. Gobasti biljard ima to lepo lastnost, da je fazni prostor ostro ločen v natanko eno regularno in eno kaotično komponento. S spreminjanjem ustreznega parametra razvrstimo stanja v regularna in kaotična brez računanja Wignerjevih funkcij. Najprej se osredotočimo na porazdelitev po razmikih med sesednjimi nivoji z namenom testiranja modela naključnih matrik, potem pa na izognjena križanja med regularnimi in kaotičnimi stanji. Razcep v izognjenem križanju nam pove o velikosti dinamičnega tuneliranja in posledično o stabilnosti določenega regularnega stanja. Večji razcep pomeni večje tuneliranje. Z upoštevanjem Fermijevega zlatega pravila smo dobili koeficiente dinamičnega tuneliranja za vsako regularno stanje gobastega biljarda. Primerjamo rezultate mikrovalovnega ekperimenta in numeričega računa z razširjeno metodo integriranja po robu z že obstoječo analitično teorijo Ketzmericka in sodelavcev. Našli smo zelo dobro ujemanje v dosegljivem območju nizkih energij. Nazadnje obravnavamo ansamble realnih simetričnih matrik dimenzij N = 2 do N = oo za različne porazdelitve matričnih elementov. Za N = 2 imamo eksaktne analitične rezultate Grossmanna in Robnika, medtem ko imamo zaiV=co teorijo Hackenbroicha in Weidenmüllerja (HW). Ta pravi, da so lokalne fluktuacije spektrov natančno popisane z GOE, če je limitna porazdelitev lastnih vrednosti gladka in omejena na končen interval. Numerično pokažemo, daje tak prehod k univerzalnemu obnašanju, ki ga napoveduje HW, zelo hiter in da se dejansko ne zgodi, če ena ali druga predpostavka v HW ni izpoljena. To testiramo za škatlasto, eksponentno, Cauchy-Lorentzovo in singularno (potenčna pomnožimo z eksponentno) porazdelitev matričnih elementov.
iv
PACS števila:
02.50.Cw Verjetnostna teorija
02.70.Pt Metoda integriranja po robu
03.65.Sq Semiklasične teorije in aplikacije v kvantni mehaniki
03.65.Xp Tuneliranje, čas prečenja, kvantna Ženo dinamika
05.45.-a Nelinearna dinamika in nelinearni dinamični sistemi
05.45.Mt Kvantni kaos; semiklasične metode
05.45.Pq Numerične simulacije kaotičnih sistemov
Ključne besede: dinamično tuneliranje, regularno, kaotično, sistem mešanega tipa, energijski nivoji, porazdelitev po razmikih med sosednjimi nivoji, dinamika stanj, gobasti biljard, izognjena križanja, koeficienti tuneliranja, teorija naključnih matrik, mikrovalovni eksperimenti
v
Acknowledgement
I would like to thank my supervisor Marko Robnik for all his effort, organisation and support during all the years of my research presented in this thesis. I would also like to thank Hans-Ju¨rgen St¨ockmann for his cooperation, great ideas and remarks. I am also grateful to Ulrich Kuhl for his great and invaluable help in the experiments and for many answers to my numerous questions.
I would also like to thank Marko Vraniˇcar, Igor Mozetiˇc and Valery Romanovski for their unselfish help at CAMTP.
I am also grateful to Roland Ketzmerich and his team from Dresden, Helena M. David and Ruven H¨ohmann for their excellent cooperation. I would like to express my gratitude to Tomaˇz Prosen and, especially, to Gregor Veble for all their advice and software assistance. In the end, I would like to thank Brigita Retar and Petra Puˇcnik for proofreading the English and the Slovenian part of the present thesis.
vi
vii
Zahvala
Rad bi se zahvalil mentorju Marku Robniku za ves trud, organizacijo in podporo vsa leta raziskovanja, ki je sedaj predstavljeno v tej disertaciji.
Potem bi se rad zahvalil Hans-Ju¨rgenu St¨ockmannu za zelo dobro sodelovanje, ideje in komentarje. Rad bi se zahvalil Ulrichu Kuhlu za veliko in prijazno pomoˇc v ˇcasu bivanj v Marburgu, za vso podporo pri eksperimentalnem delu in izvrednotenju podatkov ter za odgovore na veliko mojih vpraˇsanj. Rad bi se zahvalil tudi Marku Vraniˇcarju, Igorju Mozetiˇcu in Valeriju Romanovskemu za nesebiˇcno pomoˇc na CAMTP-ju ter skupini Rolanda Ketzmericka iz Dresdena, Heleni M. David in Ruvenu H¨ohmannu za dobro sodelovanje. Zahvala gre tudi Tomaˇzu Prosenu in posebej Gregorju Vebletu za koristne diskusije in programsko pomoˇc.
Na koncu bi se rad zahvalil Brigiti Retar in Petri Puˇcnik za lektoriranje angleˇskega in slovenskega dela te disertacije.
viii
ix
Dedication
I dedicate this thesis to my family.
X
xi
Posvetilo
Disertacijo posvečam svoji družini.
xii
Contents
1 Introduction 1
1.1 Billiards ..................................... 1
1.2 Quantum chaos ................................. 2
1.3 Dynamical tunneling .............................. 3
1.4 Random matrix theory and level statistics .................. 5
1.5 Non-Gaussian RMT ............................... 7
2 The distorted Berry-Robnik level spacing distribution 9
2.1 The Berry-Robnik level spacing distribution ................. 9
2.2 General distortion of the general level spacing distribution ......... 10
2.3 Gaussian distortion of the BR distribution .................. 12
2.4 Analytical studies of the distorted Berry-Robnik distribution ........ 22
2.4.1 Small S behaviour of PDABR(S) ..................... 22
2.4.2 Large S behaviour of PDABR(S) and PDTBR(S) ............. 23
2.5 Simulations with random matrices ....................... 24
2.5.1 The antenna distorted BR distribution (all-to-all couplings) . . . . 24
2.5.2 The tunneling distorted BR distribution ............... 31
2.6 Improvements on the tunneling-distorted BR distribution .......... 34
2.7 Dependence of the model on the matrix element statistics .......... 38
3 Dynamical tunnelinginmushroom billiards 41
3.1 Mushroom billiard - classical .......................... 41
3.2 Mushroom billiard - quantum ......................... 43
3.3 Microwave experiments ............................. 45
3.3.1 The similarity between QM and ED .................. 45
3.3.2 Resonances ............................... 46
3.3.3 Harmonic inversion ........................... 48
3.4 Level Statistics ................................. 50
3.4.1 Level dynamics ............................. 51
3.4.2 The level spacing distribution - comparison with the RMT prediction 53
3.4.3 Absorber in the foot .......................... 53
3.5 Expanded boundary integral method ..................... 53
3.5.1 Corners and the accuracy of EBIM .................. 57
3.5.2 Level dynamics - increasing the depth l ................ 61
3.5.3 The level spacing distribution - comparison with the RMT prediction 64
3.6 Avoided crossings ................................ 64
3.7 Tunneling rates ................................. 72
xiv
3.7.1 Fermi’s golden rule ........................... 73
3.7.2 Tunneling rates - the experiment ................... 75
3.7.3 Tunneling rates - the numerics ..................... 80
4 Numerical studies of non-Gaussian real symmetric random matrices 85
4.1 2D real symmetric random matrices ...................... 85
4.1.1 Box (uniform) distribution ....................... 85
4.1.2 Exponential distribution ........................ 86
4.1.3 Cauchy-Lorentz distribution ...................... 86
4.1.4 Singular times exponential distribution ................ 87
4.1.5 Comparison with numerics ....................... 87
4.2 Numerical calculations with higher dimensional non-Gaussian random matrices ....................................... 88
4.2.1 Box (uniform) distribution ....................... 89
4.2.2 Exponential distribution ........................ 89
4.2.3 Cauchy-Lorentz distribution ...................... 89
4.2.4 Singular times exponential distribution ................ 92
5 Summary and conclusions 95
Bibliography 98
Appendices 105
A ?1 calculation for the mushroom billiard 107
B Theoretical analysis of the tunneling rates in the mushroom billiard 111
C Numerics with particular solutions for the tunneling rates in the mushroom billiard 115
Daljˇsi slovenski povzetek 117
i Deformirana porazdelitev Berryja in Robnika po
razmikih med sosednjimi nivoji ........................ 118
ii Simulacije z nakljuˇcnimi matrikami ...................... 120
iii Gobasti biljard ................................. 123
iv Mikrovalovni eksperimenti in numeriˇcna metoda EBIM ........... 125
v Dinamika lastnih energij gobastega biljarda .................. 126
vi Izognjena kriˇzanja in tuneliranje ........................ 127
vii Numeriˇcna ˇstudija negausovskih realno simetriˇcnih matrik ......... 131
1
Chapter 1
Introduction
In classical mechanics there exist two types of motion - regular and chaotic. In an integrable system regular motion is stable quasiperiodic or periodic and takes place on the N-dimensional invariant tori in the classical phase space of 2N dimensions where N is the number of degrees of freedom. Since chaotic orbits are characterised by the exponential divergence of nearby orbits, they have positive Lyapunov exponents. The K-system is characterised by the chaotic region in the classical phase space with positive Lyapunov exponents and with a positive Liouville measure. If the system is ergodic as well, i.e. if almost any orbit can visit an arbitrarily small neighbourhood of any other point on the (2N - 1)-dimensional energy surface, we say that the system is fully chaotic. However, in nature the generic (or typical) systems are neither integrable (regular) nor fully chaotic but they are of mixed type. Their motion is regular on invariant tori for certain initial conditions and chaotic for complementary ones.
1.1 Billiards
The difference between the two types of motion can easily be demonstrated in billiards. The 2D closed billiard system is characterised by a free-point particle elastically bouncing off the boundary of a given Euclidean domain. The 2D billiard is integrable, if apart from the energy there exists another constant of motion. The invariant tori on the surface of section appear as one dimensional. All the orbits are stable. The only billiards, proven to be integrable, are the circular, the rectangular and the elliptic one. In the circular billiard there appears the angular momentum as the additional constant of motion. Thus, the invariant tori in this case can be either rational (resonant) or not, depending on the reflection angle selected. If the reflection angle is a rational multiple of ?, the torus is resonant and all the orbits on this torus are periodic. If, on the other hand, the reflection angle is an irrational multiple of ?, the orbits on the torus are quasiperiodic whereas the motion is ergodic on this torus. In the case of the semicircular billiard the second constant of motion is the absolute value of the angular momentum. In the rectangular billiard the constants of motion are the absolute values of momenta parallel to the walls of the billiard in question. The phase portrait of the elliptic billiard reminds one of the mathematical pendulum. Apart from the total energy the constant of motion is the product of the angular momenta about both foci of the ellipse.
On the other hand the Sinai billiard was the first proven fully chaotic billiard (Sinai, 1970), whose disk (a circular obstacle) is located in the rectangle. At that time it was
2
Chapter 1. Introduction
believed that the exponential divergence of orbits can only be achieved with the help of a non-convex boundary in a billiard. A few years later L. A. Bunimovich invented the stadium billiard (Bunimovich, 1974) and proved, that it is ergodic, that it has a mixing property and that it is a K system, which means that it is fully chaotic. The next fully chaotic billiard is the cardioid billiard whose chaoticity is proven in (Markarian, 1993). So far these are the only three rigourously proven chaotic billiards. At this point one should mention the family of Robnik billiards as an example of ’practically’ chaotic billiards (Robnik, 1983) for A > 1. This billiard family is described by the complex conformal map of the unit circle, w = z + \z2 which determines the boundary (for A = 12 one gets the cardioid billiard). 1 Another interesting example of chaotic billiards comes from the family of Africa billiards (Berry and Robnik, 1986) for A > 0.2, defined as a conformal map of the circle as well: w = z + Xz2 + AzVf.
The most frequent mixed-type systems are KAM systems in which fractal structures of the islands of stability coexist with the chaotic sea. The Robnik and the Africa billiards are KAM systems which are characterised by a continuous transition from an integrable to a fully chaotic billiard with the continuously changing (shape) parameter A. There are other examples of KAM billiards, from cosine billiard to annular billiard (Bohigas et al., 1993a), etc. However there exists a mixed-type billiard which does not share the complex structures of many regular islands surrounded by the chaotic sea. This billiard, which is characterised by a sharply divided phase space, is called the mushroom billiard. It was invented by L. A. Bunimovich and it was classically treated in (Bunimovich, 2001; Altmann et al., 2005, 2006; Dietz et al., 2006; Tanaka and Shudo, 2006). This thesis discusses the characteristics of the mushroom billiard and its quantum mechanical counterpart.
1.2 Quantum chaos
In quantum mechanics classical behaviour is related to various quantum objects, from wave functions to energy spectra of the corresponding counterparts, etc. The field in physics, which deals with chaos in quantum mechanics, is termed quantum chaos. One should emphasize the fact that in quantum mechanics chaos does not exist in the time domain since the Schr¨odinger equation is a linear equation. Therefore, quantum chaos was originally meant to be a study of quantum mechanics of classically chaotic systems. Some recent developments have proved that certain quantum integrable or chaotic systems, such as the spin chains which are presented in (Prosen, 1998b; Pineda and Prosen, 2007), are unique and do not have a classical limit. There arises a question of criteria distinguishing between regular and chaotic quantum systems. Apart from the Wigner or the Husimi functions of eigenstates lying in the quantum phase space, the most significant aspects are the statistical properties of the spectra of bound (Hamiltonian or Floquet) systems. These statistical properties are universal. The energy levels of integrable quantum systems are uncorrelated and possess the Poissonian statistics whereas in chaotic systems the neighbouring levels repel each other, which results in energy level repulsion.
On the basis of the so-called Principle of Uniform Semiclassical Condensation (PUSC) of Wigner functions in the true semiclassical limit one can identify two types of eigen-states in mixed-type quantum systems, i.e. those of the regular and the chaotic type
1The so-called Pascal snail curve (i.e. Pascal lima¸con) only resembles the Robnik billiard, but it is not the same as this billiard.
1.3. Dynamical tunneling
3
(Percival, 1973), depending on where they ’live’ in the quantum phase space. But in order to achieve such a clear classification and uniform extendedness (no localisation) of the Wigner functions, one should really be in the semiclassical limit: otherwise, there can emerge the so-called hierarchial states (Ketzmerick et al, 2000), ’living’ in the chaotic sea in the vicinity of the hierarchy of the regular islands. This can happen if the value of the effective Planck constant is not small enough.
Due to the formal similarity of the Schr¨odinger equation to other wave equations of physics the concept of quantum chaos can be generalized to the field of wave chaos. In these systems one may observe many similarities between wave mechanics and a suitably defined ’classical’ mechanics - ray dynamics. One should also study the morphology of the eigenfunctions and eigenmodes of the underlying wave equation. In Napoleon times there existed certain pictures named after E. F. Chladni (1756-1827). Chladni pictures (St¨ockmann, 1999) show the nodal lines of the first randomly-distributed dust on vibrating glass or metal plates; this phenomenon is shown qualitatively. Regular nodal patterns are typical for the integrable systems whereas the random ones appear in the chaotic systems of the underlying ray dynamics. To obtain more precise results people have used water surface waves, vibrating blocks, ultrasonic fields in water-filled cavities and microwave cavities. The Schr¨odinger equation of the 2D billiard system and the wave equation, which describes the electrical field of the low frequency modes in a thin microwave resonator of the same planar shape, are equivalent from the point of view of mathematical physics. The great advantage of microwave cavities is in their size which is in the range of about one meter, compared to the real implementations of quantum billiards (i.e. quantum dots, quantum wells, quantum corrals) with the range of /im or even nm (for a review see (St¨ockmann, 1999)). In the last fifteen years microwave cavities have become the main experimental research tool in the field of quantum chaos. In this research the cavity is mushroom billiard-shaped.
1.3 Dynamical tunneling
Classically, individual regions of regular and chaotic motion are separated from one other. However, from the perpective of quantum mechanics they are coupled by tunneling. This process has been termed ’dynamical tunneling’ (Davis and Heller, 1981) since it occurs across a dynamically generated barrier in phase space. The dynamical tunneling effect differs greatly from the majority of the familiar cases of quantum mechanical tunneling which involve tunneling through a classically forbidden region between two separate regions of coordinate space. The EBK (Einstein-Brillouin-Keller) quantized tori do not intersect in phase space, but their projections onto the coordinate space usually overlap, so there is no spatial separation of the two wavefunctions participating in the dynamical tunneling process. One can remark that if one treats the problem strictly mathematically, one can notice that such dynamically forbidden regions are connected by complex classical trajectories. Of course, complex algebra is needed in this case. Although the tunneling phenomena have been systematically studied in the context of quantum chaos for more than twenty years, we still have not managed to explain them completely.
Dynamical tunneling was first discussed in (Davis and Heller, 1981) where the authors treated the 2D anharmonic model potential and its eigenstates and they also tried to explain certain splittings in the vibrational states of molecules. They also showed that
4
Chapter 1. Introduction
symmetry breaking increases the splittings of the doublets. A few years later some important general considerations of the tunneling between the tori in phase space (Wilkinson, 1986) and the narrowly avoided crossings (Wilkinson, 1987) have been presented. Dynamical tunneling between symmetry related regular regions has been studied in many systems: in a driven anharmonic oscillator and in a driven double well oscillator (Lin and Ballentine, 1990, 1992), in coupled quartic oscillators (Bohigas et al., 1990, 1993b; Tomsovic and Ullmo, 1994; Leyvraz and Ullmo, 1996), in annular billiard (Doron and Frischat, 1995; Frischat and Doron, 1998; Dembowski et al., 2000), in certain experiments with cold atoms (Steck et al., 2001; Hensinger et al., 2001; Mouchet et al., 2001), in the time periodic kicked Harper system (Brodier et al., 2001, 2002; Eltschka and Schlagheck, 2005), etc. In such systems there exist the so-called quasi-modes (Arnold, 1989) lying on the symmetric tori in phase space.
Quasi-modes are wavefunctions which are constructed on a single torus. These functions fulfill the Schr¨odinger equation and would degenerate if tunneling was not present. But they are not eigenstates because they do not fulfill symmetry claims, ’real’ eigenstates are constructed via the linear combination of the quasi-modes lying on symmetry related tori. Therefore, if a quasi-mode, which is constructed on one of the tori, develops for a long period of time, it eventually evolves into its symmetric partner whereas its classical trajectories forever remain trapped on a single torus. While quasi-modes also emerge within the familiar 1D tunneling across the barrier, they are often surrounded by the chaotic sea in the context of dynamical tunneling. Thus, the presence of chaotic states increases the tunneling rates and causes an extreme sensitivity to external parameter variation. This process is called chaos-assisted tunneling (CAT). The tunneling, thus, annihilates the degeneracies and creates the splittings. Energy splitting is semiclassically given by
?E = A exp(-S/h) = A exp -- pdq , (1.1)
" B
where A is a constant associated with the energy of a quasi-mode, which weakly and mostly algebraically depends on h, whereas S is the classical action for the trajectories from the ’forbidden’ region along the path B. A special role in this semiclassical treatment is performed by the tori, fulfilling the EBK quantisation conditions, where the action integrals calculated for the N independent closed paths Cj on the torus are given by
Jj = f Pdq= 2vrfr n, + -4 . (1.2)
Here nj is the quantum number, j = 1...N where iV is the number of degrees of freedom of the system whereas v, is the Maslov index which counts the number of caustics encountered on the path Cj. Thus, in the far semiclassical limit, where the actions are very large, the level splittings decrease, which reduces the tunneling between the two symmetric EBK levels (1.2) according to (1.1).
For the sake of clarity one should emphasise that without any chaos involved, there always exists a direct tunneling between symmetry related quasi-modes. But if chaos is present, tunneling is usually much stronger and this is the reason why CAT is so important. We can see this clearly if the rate of chaoticity is treated as a perturbation, as it can be observed in the case of the model of two coupled quartic oscillators in (Tomsovic
1.4. Random matrix theory and level statistics
5
and Ullmo, 1994). Direct tunneling is preferred for smaller couplings whereas CAT is preferred in the case of larger couplings. In this process, which can be best described as a 3-level mechanism, two regular states (the doublet) and one chaotic state (which is not part of the doublet) function as the media in the neigbourhood of the two quasi-degeneracies. The reason why CAT is often preferred to direct tunneling lies in the fact that chaotic orbits from the chaotic region between two tori connect these tori classically and, thus, this type of classical transport supports quantum tunneling.
Tunneling rates also increase in the case of resonance-assisted tunneling. This type of tunneling occurs if in regular islands the complex KAM structure of resonant tori becomes important (Brodier et al., 2001, 2002; Eltschka and Schlagheck, 2005).
Dynamical tunneling has also been studied from a single regular region to the chaotic sea: in the kicked rotor (Hanson et al., 1984), in the kicked rotors with complex classical dynamics (Shudo and Ikeda, 1995) and via localisation suppression of the tunneling effects (Ishikawa et al., 2007), in the ionisation of hydrogen atoms in polarized microwaves (Za-krzewski et al., 1998), during the decay of quantum accelerator modes (Sheinman et al., 2006), in electronic quantum transport through nanowires (Feist et al., 2006), etc. Tunneling is essential whenever two eigenvalues, one from the chaotic and one from the regular region, are degenerate. Due to tunneling there emerges a quantum-mechanical mixture of the respective eigenfunctions ?C and ?R, so the degeneracy vanishes. This type of dynamical tunneling is widely discussed in the present thesis, but also CAT emerges in a slightly different way from the one presented above.
Unlike the 1D tunneling through a barrier, it is extremely difficult to quantitatively predict dynamical tunneling. Results have only been found for specific systems or system classes so far, most recently for 2D quantum maps. The latter has been studied with the help of a fictitious integrable system (B¨acker et al., 2008b). However, a precise knowledge of tunneling rates is of great importance. Apart from our spectral statistics model in systems characterised by the mixed phase space recently studied examples are the eigen-states affected by the flooding of regular islands (B¨acker et al., 2005, 2007) and emission properties of optical micro-cavities (Wiersig and Hentschel, 2008). So far the dynamical tunneling rates have not been fully quantitatively predicted since they require fitting by the factor of about 6 in the case of the annular billiard (Frischat and Doron, 1998) and by the factor of about 100 in the case of the mushroom billiard (Barnett and Betcke, 2007).
In Chap. 3 we have presented a combined experimental, theoretical, and numerical investigation into the tunneling rates in mushroom billiards. The tunneling rates were calculated with the help of Fermi’s golden rule of time perturbation theory applying to the avoided crossings between the regular and the chaotic eigenstates. The most important results have been published in (B¨acker et al., 2008a). Although the theoretical and one of the numerical parts of the research have not been provided by the author of the present thesis they are also briefly presented and discussed for the sake of completeness. Another significant topic is avoided-crossing distribution for the mushroom billiard (which is presented in Chap. 3).
1.4 Random matrix theory and level statistics
Random matrix theory (RMT) (Mehta, 1991; Guhr et al., 1998) has important applications in many branches of physics such as elementary particle physics, nuclear physics,
6
Chapter 1. Introduction
atomic physics, molecular physics, solid state physics, and, especially, in quantum wells (Narimanov and Stone, 1999). In quantum chaos RMT has proved to be an excellent model for the statistical properties of energy spectra of chaotic Hamiltonian systems. It goes back to the original paper (Casati et al., 1980) and to the classic paper (Bohigas et al., 1984) where the conjecture about the universality of RMT for classically fully chaotic systems was formulated. The conjecture has been theoretically supported by the dynamical and the semiclassical theory of spectral rigidity (Berry, 1985). The next important step beyond this approximation has been achieved in (Sieber and Richter, 2001), followed by the development of an expanded semiclassical theory (Mu¨ller et al., 2004; Heusler et al., 2004; Mu¨ller et al., 2004).
Classically integrable quantum systems show the Poissonian statistics, which is well known and corroborated by numerous analytical and numerical works (Berry and Tabor, 1977; Robnik and Veble, 1998). In chaotic systems spectral fluctuations obey the statistics of the Gaussian orthogonal (GOE), the unitary (GUE) or the simplectic (GSE) ensembles (depending on the existence of antiunitary symmetry (Robnik, 1986) and on an internal degree of freedom - spin). Usually the well known Wigner surmise, Eq. (2.4), well describes the level spacing distribution for the infinite dimensional GOE level spacing distribution. For higher-order level spacing distributions of chaotic systems useful approximate closed form formulae have been derived in (Abul-Magd and Simbel, 2000).
The spectral statistics in mixed-type systems rests upon the PUSC of the Wigner functions, upon the theoretical foundations in (Berry, 1977), and upon the theory of superposition of the so-called E(k,L) statistics (Berry and Robnik, 1984). E(k,L) statistics is the probability of having exactly k levels in an interval of the length L after unfolding, where the level spacing distribution P(S) is the second derivative of the gap probability E(0,S). The theory states that the E(k,L) statistics factorise in the strict semiclassical limit of a sufficiently small effective Planck constant heS, where the factorisation is a direct consequence of the statistical independence of the sequences of the regular and the irregular levels.
The Berry-Robnik (BR) theory and its resulting formulae have been verified in many different systems in the asymptotic semiclassical regime: in the ’far’ semiclassical limit of quantised standard map on the torus (Prosen and Robnik, 1994a,b), in the periodically pulsed spin system (Jacquod and Amiet, 1995), in the 2D semiseparable oscillator (Prosen, 1995), in the quartic generic KAM billiard with the border described by r(0) = 1 + a cos(40) with the simple phase space structures reflected in the lower transition point (Prosen, 1998a), in the E(k, L) statistics for the standard map on the torus and the quartic billiard (Prosen and Robnik, 1999), in the E(k,L) statistics with a generalised Wigner surmise tested on the Henon-Hiles potential (Abul-Magd and Simbel, 2000), in the system with a sharply divided phase space of kicked-rotor-type map in the ’near’ semiclassical regime (Malovrh and Prosen, 2002), in the Andreev billiard (Kaufmann et al., 2006), etc. There have been a number of attempts in literature to describe correctly the level statistics in the mixed-type systems and all these approaches interpolate between the RMT and the Poissonian statistics, but, unlike the BR, they are not based on sound physical grounds, and they have not been confirmed in the semiclassical limit. For a review see Chapter 3.2.2 of (St¨ockmann, 1999).
If the semiclassical limit is not reached, one observes deviations from the BR behaviour which emerge due to both the localisation effects and the tunneling between the regular and the chaotic regions in the quantum phase space of the Wigner functions of the eigen-
1.5. Non-Gaussian RMT
7
states. This results in the linear behaviour of P(S) at small S, as predicted qualitatively in (Berry and Robnik, 1984), and in the fractional power law level repulsion (which was first presented in (Prosen and Robnik, 1994b)) for S between the linear level repulsion regime and the BR tail. Presumably, the main reason for the fractional power law level repulsion lies in the localisation effects (for more details see (Izrailev, 1989, 1990)).
Here, we would like to offer a new random matrix model in order to generalise the BR level spacing distribution by including the tunneling effects between eigenstates. We will not consider localisation effects, except for the special case where p can be replaced by some effective pe//; for example in the case of flooding (B¨acker et al., 2005, 2007) or in the case of hierarchical states (Ketzmerick et al., 2000). So, we will assume that the PUSC is fulfilled and the level splittings are affected by tunneling mechanisms which couple two regular and two chaotic levels and also regular-chaotic levels through the chaotic levels functioning as the intermediary. Thus, our purpose is to mimic tunneling by means of a two-level random matrix model where the non-diagonal elements correspond to the tunneling matrix elements.
Following some recent advances in the non-Gaussian and the non-normal random matrices in (Grossmann and Robnik, 2007a,b) we will try to verify to what extent the properties of the matrix ensemble remain structurally stable, i.e. robust, against certain variations of the model properties such as the statistics of the matrix elements. Finally, we will use the theoretical results to describe the level spacing distribution of the eigenvalues of a various configurations of the mushroom billiard, both on the basis of the experimental data for the microwave cavities as well as on the basis of the numerical data obtained by the expanded boundary integral method.
This work, published in (Vidmar et al., 2007), is presented in Chap. 2 and, partially, in Chap. 3 of the present thesis.
1.5 Non-Gaussian RMT
The last part (Chap. 4) of the thesis, also presented in (Robnik et al., 2007), contains the eigenvalue calculations for different random matrix ensembles where the distribution of the matrix elements is non-Gaussian. The importance of the Gaussian ensembles of the RMT has been corroborated by theory in (Hackenbroich and Weidenmu¨ller, 1995), where the authors have considered other random matrix ensembles than the Gaussian ones (GOE, GUE and GSE). They have proved (using the supersymmetric techniques) that the local spectral fluctuations (after spectral unfolding) obey universal statistical laws described by the Gaussian ensembles in the limit of the infinite dimension N › ?, independent of the distribution of the matrix elements, provided that two conditions are fulfilled: (i) the energy level distribution function in this limit should be smooth, and (ii) it should be confined to a finite interval. Thus, this important finding is some kind of a central limit theorem for all the spectral fluctuations of random matrices.
Usually the renowned Wigner formula (Mehta, 1991) well describes the infinite dimensional GOE (or also GUE and GSE) level spacing distribution. Recently in (Grossmann and Robnik, 2007a) the authors have also studied 2D non-normal Gaussian matrices (with asymmetric variance) and a number of non-Gaussian ensembles of 2D real symmetric random matrices in (Grossmann and Robnik, 2007b) where explicit analytic results for level spacing distribution have been obtained. It has been shown there that the level repulsion
8
Chapter 1. Introduction
is a very robust phenomenon, which only depends on the behaviour of the distribution of the matrix elements, i.e. on its regularity at zero value. If the distribution functions of the diagonal and the off-diagonal matrix elements are regular and nonzero at zero value, there always appears the linear level repulsion. If the distribution functions in question are regular and at least one of them is zero at zero value, then the level repulsion is quadratic. If the distribution of the matrix elements is singular at zero value (e.g. an integrable power law times exponential), there appears the fractional power law level repulsion (Prosen and Robnik, 1994b; Prosen, 1995). The Cauchy-Lorentz distribution is also interesting, since it does not contain a finite first moment and produces P(S) which is not normalisable to the unit first moment (mean level spacings).
It is straightforward, interesting and important to generalise these analytic 2D results to the higher dimensional case N > 2 where analytical results are practically impossible. Our extensive numerical calculations have shown that for regular matrix ensembles there applies the HW theorem. In the last two singular cases (mentioned earlier) all the assumptions from HW theory are not fulfilled, i.e. condition (ii) is not realised and, due to this, the systems strongly violate the GOE behaviour permitted by the HW theorem which is presented in Chap. 4.
9
Chapter 2
The distorted Berry-Robnik level spacing distribution
2.1 The Berry-Robnik level spacing distribution
Gap probability E(0,S) is the probability that there are no levels on the interval of the length S after spectral unfolding. The level spacing distribution P(S) for an unfolded spectrum is equal to the second derivative of gap probability: P(S) = d2E(0, S)/dS2. Gap probability factorises upon a statistically independent superposition of independent level sequences. From here onwards gap probability is denoted by E(S). The basic finding in the Berry-Robnik (BR) picture (Berry and Robnik, 1984) is that the E(S) for the total spectrum of a mixed-type system is the product of all the gap probabilities whose arguments must be weighted by the classical parameters Pi measuring the relative volume of the phase space of the regular component (i = 1), of the largest chaotic component i = 2, of the next largest chaotic component (i = 3), etc. We have focused on the simple (and usually sufficient) approximation of only two components, i.e. the regular component i = 1, with P1 = p, and the chaotic component one i = 2, with p2 = 1 - p, so that p1+p2 = 1. Thus, gap probability for the entire spectrum equals E(S) = E1(p1S)E2(p2S). The BR level spacing distribution can be presented by
Pbr(S) = ^2 E1(P1S) E2(p2S). (2.1)
Here we have introduced two quantities related to E(S). The first one, F(S), indicates the probability that the level spacing is larger than S. The second one is the cumulative level spacing distribution which equals: W(S) = 0 P(x)dx. The relations between two quantities can be represented as: F(S) = -dE/dS = 1 - W(S). Now the BR level spacing distribution (2.1) can be rewritten thus
Pbr(S) = p21P1E2 + 2p1p2F1F2 + p22E1P2 , (2.2)
where the argument of each quantity with the index % equals Xi = PiS, % = 1, 2. For the chaotic case we have to apply the GOE (GUE, GSE) results for infinite matrices, for which there exist no closed form expressions. For the analytical case the Wigner distribution, which is an exact GOE result for the 2 x 2 matrices (while the GUE and the GSE are not treated here), is often used as an excellent approximation. The discrepancy between the
10 Chapter 2. The distorted Berry-Robnik level spacing distribution
Wigner distribution and the exact result for the infinite matrices is only up to 2%, so the Wigner approximation is suitable for many practical and, especially, for many analytical purposes. In the regular case, i = 1, we will use the Poissonian statistics (Berry and Tabor, 1977)
E1(x1) = F1(x1) = P1(x1) = e-x\ (2.3)
whereas in the chaotic case, % = 2, we will employ the Wigner (2D GOE) approximation; see (Mehta, 1991).
P2(x2) = — exp i-^4) , ^(r2) = 1 - W2(x2) = exp (~^4) , (2.4)
and
L2(2:2) = 1-erf y^A =erfcy^A (2.5)
where erf (x) = -^ /0X e~u2 du is the error integral whereas erfc(x) = 1 - erf (x) represents its complement. With (2.3) - (2.5) the BR level spacing distribution function (2.2) can be written as (Berry and Robnik, 1984)
Pbr(S) = e~plS ie- 22 (2Plp2 + ^L^\ + p?erfc (^f^] X . (2.6)
FLF 2 ,l 2
The PBR and its first moment are normalised. The second moment (S2) = 2 j™ E(S) dS can sometimes be useful as well. For the BR level spacing distribution the second moment can be expressed in a closed form
{2 \
1 - e 2 erfc ( -^ ] \
(S 2)br = - 1 - e2erfc-^ . (2.7)
Pi
In case of the Poissonian statistics (Pl = 1, p2 = 0) (S2) equals 2, while in the case of the Wigner distribution (Pl = 0, p2 = 1) {S2} equals 4/vr.
2.2 General distortion of the general level spacing distribution
Let us consider 2x2 real symmetric matrices A = (Ay) where i,j = 1 or 2. Only the difference between the eigenvalues is of relevance in the present context. Without any loss of generality (Grossmann and Robnik, 2007a) one may assume that the trace of A vanishes
A= ( ? M , (2.8)
o —a
where a and b are real. Thus, for the eigenvalues of A one obtains
Ai,2 = ±Va2 + b2 , (2.9)
2.2. General distortion of the general level spacing distribution
11
where the difference A1 - A2 between the both eigenvalues equals 2y/a2 + b2. Thus, the level spacing distribution is given by
+00 +(»
/ / da db 5 LS - 2V
—00 —00
P(S)= dad&čS-2a2 + &2U„(a)#fc(&), (2.10)
—00 —00
where 5(a;) is the Dirac delta function whereas #0(a) and #,(&) are the normalised probability densities of the diagonal and the off-diagonal matrix elements a and b respectively. In the above construction P(S) is defined for S > 0. P(S) is automatically normalised,
+00 +OO oo
00 +oo +oo oo _________
/ P(S) dS = / / / da db dS 5 LS - 2Va2 + b2\ ^a(a) ^6(6)
0 —00 —00 0
^a(a) da / ^6(6) d6 / 5 (^ - 2Va2 + b2) dS
0
= 1 , (2.11)
but this is not true for its first moment; in general /0°° S P(S) dS=1. If we now introduce the polar coordinates
a = rcos^, 6 = rsin^ (2.12)
where r G [0,oo) and 0 < \ p = 0.50 (j = 0.04 -
0.6 —
0.4 N -
0.2 - -
o n .........i ........i..... . . . TTTTTt.....
Figure 2.6: The same as Figs. 2.1 - 2.5 presenting the regular fraction p = 0.5 and various values of the level coupling parameter a = 0.01,0.02,0.03,0.04.
This is, in fact, the weighted mean of the undistorted and the antenna-distorted (distorted by the all-to-all couplings) BR distributions. For both limiting cases p = 0 and p = 1 the original GOE and PE (the Poissonian ensemble) level spacing distributions are recovered as expected. One should note that the tunneling-distorted BR distribution (2.22) is normalised to unity whereas its first moment (i.e. the mean level spacing) is not. In order to compare the theoretical level spacing distribution with the level spacing distribution from the real data (experimental or numerical ones) the rescaling, or the normalisation of the first moment to one, has to be performed exactly as suggested in Eq. (2.20). In the end we obtain the final theoretical tunneling-distorted BR distribution function by using the rescaling factor BT
pdbr(s) = bt PDbr(btS)} with BT
o
X
PlBR(x) dx.
(2.23)
The BTs for the set of parameters (p, a) are shown in the forth column of Tab. 2.1. They appear in the range [1,1.1] for the parameters selected.
Again, for each set of parameters (p, a) the integrals in Eqs. (2.23) and (2.17) are evaluated numerically The resulting P%%R(S)s are plotted in Figs. 2.1 - 2.6 with the full red curve. One can clearly see that there are no examples of repulsion. Why does this happen? For mathematical reasons PE'BR(S) is a superposition of PßBR(S) and PBr(S) where the latter has this property This feature is discussed in more detail in the Subsec. 2.5.2. At larger values of S of the distribution PlBR(S) one also notices the overshooting and,
_
22 Chapter 2. The distorted Berry-Robnik level spacing distribution
asymptotically, Pdbr(S) also behaves like the stretched BR distribution.
2.4 Analytical studies of the distorted Berry-Robnik distribution
We will calculate analytical approximations for the antenna-distorted BR level spacing distribution PßBR given in Eq. (2.17), i.e. for small S and for large S, by using (2.6) for Per in the general expression (2.17).
2.4.1 Small S behaviour of PßBR(S)
Let us expand PBR(x) from (2.6) into the Taylor series at small x,
oo
Pbr(x) = ^akxk. (2.24)
fc=0
Then PLBR(S) is given by the series, from Eqs. (2.17) and (2.24),
Pdbr(S) = ^^Y,akS k r/2 coSk
/ X K
7T/2 /xTf + 1 -. 7
cosfc (/? exp (-asm2 1. The main contribution to the integral comes from the interval close to 14 all the results are practically the same. We have also tested matrices with N = 50 (and smaller) and observed a very good agreement as well. Therefore, we believe that the results of the model do not depend on the dimension of the matrices, so, for instance, the option N = 100 and unf = 20 would be equally or (even) slightly more appropriate, as indicated in Tab. 2.2.
We have also tested the phenomenological rule with unf (for N = 1000) by using the GOE random matrices (p = 0, a = 0) where the optimal agreement with the exact unfolding (using the Wigner semicircle rule) has been sought. The results confirm the ones obtained from the X2 test, which means that unf = 30 is the best possible choice in this case where the unf does not influence the results to such a large extent.
If we consider the results of PDABR(S) and PDA BnRN(S) for sufficiently small values of a, we notice one interesting feature: smaller spacings are affected to much a larger extent
2.5. Simulations with random matrices
31
than bigger ones (for the non-normalised distributions the effect would be even greater). For the 2D model this can be explained in the following way: the spacing S = En+l - En of the coupled system, which is expressed in terms of the spacing of the uncoupled system So = En+i,o - En,o = 2a > 0 as well as in terms of the coupling b (for the definition of matrix elements a and b see Eq. (2.8)), is given by
S=Sl + 4b2 (2.41)
If Enfl and Lra+i)0 are equal (i.e. a = 0), then S = 2b, otherwise S = S01 + ^L. If a is
small, which we have assumed here, the coupling b is small as well, i.e. b < SQ. Thus, we can only take the first order in the Taylor expansion and get
S =So + 2^. (2.42)
The So from the denominator explains why smaller spacings change to much a larger extent than bigger ones. It seems that the two-level process dominates in the iV-dimensional model as well.
2.5.2 The tunneling distorted BR distribution
In order to describe the tunneling between the regular and the chaotic phase space regions in physical systems we will assume that the non-zero off-diagonal matrix elements only occur between the regular and the chaotic states but not within the regular or the chaotic block itself. In the cross coupling case the two-level approximation expresses the tunneling-distorted Berry-Robnik formula P^R(S) which is a weighted mean of the antenna-distorted and the undistorted Berry-Robnik formula which is normalised to (S) = 1, as presented in Eqs. (2.22) and (2.23). Fig. 2.11 shows the results from the simulations of PE'brn(s) and from the two-level expression for PlBR(S), for a = 0.05 (as also applied for the all-to-all couplings) and for a number of p values. Fig. 2.12 represents PE'brn(s) and the PEbr(S) for a = 0.1 and for a number of p values. Fig. 2.13 displays the results for p = 0.5 and a = 0.01,0.02,0.03,0.04. In all the cases the agreement between the iV-level simulation and the two-level analytical formula (2.22) is rather encouraging. The agreement is better for larger values of p where the regular part dominates.
Although the general agreement for a = 0.05 is still good, there appear significant deviations between the two-level approximation and the results from the N-dimensional simulations for small values of S. In the iV-dimensional simulation the linear level repulsion is always observed in cases of small values of S while the two-level approximation starts with a non-zero value at S = 0. For a steep linear level repulsion the region in S, which should appear within an exponentially small interval oc exp(-const./heff) typical of tunneling phenomena, is indeed very small (see the insets in Figs. 2.11 - 2.13). This reflects that the accidental degeneracies, which occur generically in the regular part of the spectrum, are not lifted in the case of the two-level approximation. In the N-dimensional simulations, on the other hand, all the degeneracies are lifted in the regular block although there are no direct tunneling matrix elements; there only appears the second order tunneling which couples two regular states indirectly, i.e. via one or more chaotic ones. Here the two-level approximation is certainly not appropriate. 3
3Here the term ’degeneracies’ also includes nearby lying states, i.e. the so-called quasi-degeneracies.
32 Chapter 2. The distorted Berry-Robnik level spacing distribution
Figure 2.11: The numerical histogram for the N-level (N = 1000) tunneling distorted BR level spacing distribution P]j%RN(S) is compared with the pEbr(s) from the 2-level model (2.23) for the fixed coupling parameter a = 0.05 and for various sizes of the regular regions p = 0.10,0.25,0.35,0.50,0.75,0.90. For the sake of comparison the blue dashed line represents the BR distribution with the same p. Only the couplings between the regular and the chaotic eigenvalues are allowed in this case; couplings cannot occur between two regular states or two chaotic ones. The insets show the small S behaviour of all the three curves plotted. The tunneling formula (2.23) is in good agreement with the numerical simulations. For the lower left plot with p =0.75 there are 10 times more objects in the statistics.
We should also mention that neither localisation (Leyvraz and Ullmo, 1996; Ishikawa et al., 2007) nor flooding effects (B¨acker et al., 2005, 2007) have been treated so far. Typically, localisation of chaotic states (as a deviation from the PUCS-uniformity) would
2.5. Simulations with random matrices
33
1.0
1.0
1.0
0 8 P ö = 0.25 = 0.10
0.6 "f\ " 1 -
0.4 K V °' -j -
0.2 -
o n ....................
Figure 2.12: The same as in Fig. 2.11 but for the doubled cross-coupling (tunneling) strength a = 0.10 and for various values of the relative size of the regular regions p = 0.10,0.25,0.35,0.50,0.75,0.90. Again only cross couplings of the integrable with the chaotic states are allowed here; couplings cannot occur between two regular states or two chaotic ones. All the plots have the same number of objects in the statistics.
be modelled by the suppression of tunneling between certain basis-states. Localisation near regular (stability) islands (classically stickiness) would be modelled by replacing ? with some larger effective ?eff, whereas (partial) flooding of regular (stability) islands would be modelled by some smaller effective ?eff.
34 Chapter 2. The distorted Berry-Robnik level spacing distribution
0.0
1.U ......... p = 0.50 a = 0.02
n R
\ I - f J^aV^Ir^M,_
0.6 1 \l If f -
0.4 >t I - -
0.2 n n .................... , , TTTtt......
0.0
Figure 2.13: The same comparison of level spacing distributions as presented in Figs. 2.11 and 2.12, but now for the fixed ? = 0.5 (equal for both the regular and the chaotic parts) and for other (smaller and larger) coupling ? = 0.01, 0.02, 0.03, 0.04, 0.20, 0.30 affected by tunneling. Again only cross couplings between the integrable and the chaotic eigenvalues are allowed. In the upper left plot with ? = 0.01 there appear 10 times more objects in the statistics.
2.6 Improvements on the tunneling-distorted BR distribution
Sec. 2.5.2 proves that the 2D model has certain disadvantages. This section represents two possible improvements to this model.
The formulation from Eq. (2.21), which presents the distribution of tunneling matrix elements, could be replaced by
2.6. Improvements on the tunneling-distorted BR distribution
35
gb(b) = 2p(1-p)—1=exp—b— +[1-2p(1-p)]—1=exp—^ , (2.43)
where 0 phenomenologically represents the second order splitting in the regular block. This block is not fully regular here, so it is not proportional to 5(b). Since the splitting of the regular states is the second-order effect, ox should be interpreted as being proportional to a2. So far we have not been able to establish an explicit relation between ox and a in this modification of the model. This is open for the future research.
To remove the degeneracies appearing due to the 2D model, one could use the 3 x 3 matrices, as proposed in (St¨ockmann, 2007). These are chosen in such a way that they indicate the tunneling which couples two neighbouring regular states with the closest chaotic state: i.e. R-C-R tunneling. Since this coupling causes the repulsion of the regular levels, it also removes the degeneracies appearing in the regular sequence.
At this point we will select a mixed-type sequence on the basis of which we will develop the 3-level theory. Our Hamiltonian is written as
EnR 0 Hn= | 0 E{n+1)R
Win W2n
Wln \ EnR + E{n+l)R ( 5" 0X
W2n |---------------^----- 1 = | 0 — On
2
Ec Win W2n
Win
W2n I (2.44)
?n
where n is the index of the regular levels whereas EnR, E{n+i)R are the neighbouring regular levels with EnR < E{n+i)R and Ec is the closest chaotic level. wln and w2n are couplings of one regular and one chaotic level whereas 5n = EnR~E( n+vR and ?n = Ec- gnR+^(n+DR. The characteristic polynomial is thus expressed as
X(E) = det(Hn -E) = (E2 - 52n)(E - ?n) - (w\n + w22n)E + (w\n - w22n)5n. (2.45)
The chaotic level can lie above or below the mean of two neighbouring regular levels. Therefore ?n may have positive or negative values with equal probability and the mean (?n}n (the average is calculated for all the regular levels) equals zero in this case. In addition, the averages (w2ln)n = w\ and (w22n)n = w\ are equal.
Rigorously, this has to be applied to all the chaotic states affecting all the regular pairs. But, as presented at the end of Subsec. 2.5.1, the states that are far away from one another do not influence each other. In the first approximation we have therefore selected two neighbouring regular levels influenced by a close lying chaotic one. At this point we can make two additional approximations. Thus, we replace:
a) ?n with its mean (?n)n = 0. This indicates that the closest chaotic level lies in the very middle of two regular levels. This is not completely true since the chaotic levels would then obey the Poissonian statistics as well. Despite this the results obtained in this way could be useful.
b) wjn - w\n with its mean (wjn - w22n)n, which results in (w2ln)n - {wln)n = w2-w% = 0. The final result will show whether the two approximations are meaningful.
By including the approximations into Eq. 2.45 we obtain
X(E) = (E2 -6n- w\n - w22n)E,
(2.46)
36 Chapter 2. The distorted Berry-Robnik level spacing distribution
with the solutions
K'" = ±\/ön + w21n + w22n, Ei11 = 0. (2.47)
Thus, the chaotic state does not change in any way whereas certain changes can be observed in the case of the two regular states. The distance between the latter equals
AEn = \Ei - E^\ = 2yJ52 + w2n + w2n. (2.48)
At this point we would like to learn how the regular spectrum changes if one follows the above procedure. For the spacing distribution of the regular levels PPoissore(:z1), where x1 = 2\6n\, we have chosen the distribution from Eq. (2.3), weighted by the fraction p of the Nreg regular levels appearing in the spectrum
Ppoissan{X1) = P GXp{-PX1). (2.49)
The new distribution PDPoissm(x), which is termed the distorted Poisson distribution, reads
1 N re9
9 n=1
1
IV reg i / _________________________\ \
^28 [x- 2J82n + w12n + w22n \
n=1 wln ,w 2n
iv Teg I / _______________________________\ \
^ J2 / dx1 5 (x1 - 2\5n\) (ö(x- ^ x21 + 4(w2n + w22n) J \
y n=1 \ \ / /
-, 'Veg / / __________________________\ \
/ dx1 — J26(x1- 2\6n\) /ölx- \jx21 + i(w2n + w2n) J \
(2.50)
where %) is the Dirac delta function of y whereas the average is calculated for all the off-diagonal elements w1n and w2n. By evaluating the sum over n in the limit of large Nreg we obtain
J dx1 PPotssm(x1) (ö (x - ^ x21 + 4K + w2) \ \
PDPmssan(x) = dX1 PPmssan(X1) 6X-x21 + A{w12 + W2) , (2.51)
where W1 and w2 are the average couplings described above. Here we have assumed that W1 and w2 are Gaussian distributed (2.16), which was also the case in the 2 x 2 model:
gb{w) = -1 exp (-L*) where w could either be w1 or w2, both with the same variance
a2. If we incorporate this into (2.51), we obtain
oc OO OO 2,2 / ,---------------------------------------\
/ II w~\ iw9 I \
/ dx1e-pXl / dw1 / dw2 e"^^" 5 x - \jx21 + i(w2 + w2) .
0 — oo — oo
(2.52)
PDpmSson(x) = ^2 dX1e-^ dW1 dw2e-^^5x
_
_
_
2.6. Improvements on the tunneling-distorted BR distribution
37
If we introduce the polar coordinates: w = w1 cos oo (Hackenbroich and Weidenmu¨ller, 1995), provided that the level density in this limit is smooth and confined to a finite interval. For 2D ensembles an explicit analytic theory has recently been performed (Grossmann and Robnik, 2007a,b) for a variety of matrix element distribution functions, including the exponential and the box (the uniform) distributions. Extensive numerical studies of such ensembles at high dimensions of the matrices (see Chap. 4) confirm the Hackenbroich-Weidenmu¨ller prediction and show that the transition to the universal behaviour is pretty fast. If one or both of the Hackenbroich-Weidenmu¨ller conditions are not fulfilled, e.g. for the Cauchy-Lorentz distribution of the matrix elements, there appear deviations from the GOE level spacing distribution.
Bearing in mind all this we would like to explore the sensitivity of the distorted BR level spacing distribution as regards the choice of the matrix element statistics. Thus, in addition to the Gaussian distribution treated earlier, as defined in (2.16), we will consider the box (the uniform) distribution
9b(b)
-y/3
if | 6 |< aV3 and zero, otherwise,
(2.59)
and the exponential distribution
9b(b)
2^
exp
( \b\2
(2.60)
both with the variance a2.
It is obvious that the deviations from a Gaussian distribution are not as large as one might expect, which is clearly demonstrated in Figs. 2.15, 2.16, representing the results for the normalized antenna-distorted BR distribution P^RN(S), for the Gaussian, for
_
_
40 Chapter 2. The distorted Berry-Robnik level spacing distribution
Figure 2.16: The same as Fig. 2.15 for the box (instead of the exponential) distribution of the off-diagonal matrix elements: the histogram from the numerical simulations with the matrices N = 1000 well agrees with the 2D formula (solid curve), but both of them only slightly deviate from the Gaussian model (dotted curve).
the exponential and for the box (the uniform) distribution with the same dispersion a2. These are various theoretical curves with the same a2. One should emphasise that no best curve fitting has been applied. One notices that the 2D theory for P$%R(S) from Sec. 2.3 is a suitable approximation for the random matrix ensembles of the dimension N = 1000. Nevertheless, the curves slightly differ from one another, especially in the central region of 0.3 < S < 0.6 whereas the small S behaviour (the level repulsion) and the large S behavior (the tail) are in good agreement with the model where the gb(b) is the Gaussian distribution function. In fact, the tail for the exponential distribution is, analytically, the same as one for the Gaussian model (2.37). The ratio of the slopes at S = 0 for the exponential and the Gaussian case equals yfii whereas this ration equals V^/6 in the case of the box and the Gaussian distribution. This ration remains the same within 2% after one has rescaled the first moment to unity by using the stretch factor BA. The agreement in the case of the Gaussian and the other ensembles is, therefore, quite satisfactory.
One intuitively expects that the statistical spectral properties of random matrix ensembles mostly depend on the variance of the matrix elements and not on other details of the ensemble unless these become singular in one way or another.
Some basic results from this chapter are tested in the following one.
41
Chapter 3
Dynamical tunneling in mushroom billiards
3.1 Mushroom billiard - classical
The mushroom billiard (shown in Fig. 3.1) was introduced in (Bunimovich, 2001). It is composed of a semicircle - the head, and a rectangle - the foot. We have used the rectangle for the foot although some other shapes, such as a triangle, etc., may be used instead. The mushroom billiard is a 2D autonomous system with the Hamiltonian H(p,q) = p2/2M + V (q), where (q,p) are 2D coordinates and momenta of the particle whereas M is its mass and V is its potential. The potential is zero inside and infinite outside the billiard domain D.
d A
b
2a 1
Figure 3.1: The geometry of the mushroom billiard: R = radius of the semicircle, 2a = width of the foot, l = height of the foot, b = position of the foot (the larger distance from the edge).
The energy is the only global constant of this billiard. There exists one additional local constant of motion for the orbits which do not enter the foot, i.e. the absolute value of the angular momentum. The orbits with the same angular momentum form a semicircular caustic. There is one critical caustic which rigorously separates the orbits into the regular and the chaotic ones. The orbits inside the head with a larger or equal caustic forever remain in the head whereas the other orbits eventually enter the foot, so they are chaotic (top two of Fig. 3.2). Since the billiard system has precisely one chaotic and one integrable component in the phase space, which are well separated from one
42
Chapter 3. Dynamical tunneling in mushroom billiards
another (bottom of Fig. 3.2), this system is particularly attractive for analysis. Such a clear separation of the phase space first appeared in certain classical maps (Lee, 1989; Malovrh and Prosen, 2002).
1.0
0.5
Q- 0.0 i'
0.5 -:
1.0
o
3
5
6
Figure 3.2: Top-left: Example of a regular (70 reflections) orbit in the mushroom billiard. Top-right: Example of a chaotic (100 reflections) orbit. Bottom: Phase space portrait. The abscissa is the border coordinate s (of the reflection point) beginning at the right ?2 corner of the semicircle whereas the ordinate p is the sinus of the reflection angle.
The regular part of the phase space of the mushroom billiard is simple since it consists of a single regular island. Due to this the volume of the regular (or the chaotic) part of the phase space can be calculated much easier than in the case of a KAM system. The group from Dresden, which was lead by R. Ketzmerick, derived an analytic expression for the effective chaotic area of the mushroom billiard with the central (b = R - a) foot
3.2. Mushroom billiard - quantum
43
Ach = 2la +
R2 arcsin (|) + av^R2^2
(3.1)
a
where Ac7l is the effective chaotic area of the configuration space of the mushroom billiard. If one wants to obtain the fraction p2 of the chaotic component, the Ach has to be divided by the entire area of the billiard, i.e. A = 2 la + ^. If the foot is not in the central position (b=R-a), then instead of a in [... ] from Eq. (3.1) one employs a' defined as: a' = 2a + b-R. This calculation is represented and confirmed in Appendix A.
While the regular component of the phase space is very simple, the chaotic one consists of a complex distribution of families of marginally unstable periodic orbits (MUPOs). The first class of MUPOs corresponds to the orbits bouncing between the parallel walls in the foot of the mushroom billiard. Similar MUPOs can be found in many other billiards with parallel walls. The other, more interesting class of MUPOs, corresponds to periodic orbits in the chaotic region which remain trapped in the head of the mushroom billiard (Altmann et al., 2005, 2006). Since these sticky chaotic orbits, characterised by an almost regular motion and by the zero Lyapunov exponent, have measure zero, they do not affect the ergodicity of the system. Nevertheless, due to the MUPOs with long periods, one can understand certain dynamical properties of the system, such as the transport and the decay of correlations. As presented in (Altmann et al., 2005) these MUPOs result in an exponent 7 = 2 for the asymptotic power-law decay of the commutative recurrence time distribution Q(T) oc T~\ If P(t)dt indicates the probability that the first return to the chosen region will happen at the time between t and t + dt], NT is the number of recurrences in that region with the recurrence time t > T whereas N represents the total number of recurrences, so that then Q(T) = L~T P(t) = lim^«, ^. In fully chaotic hyperbolic systems Q(T) decays exponentially. This definitely proves that the system discussed has the property of stickiness.
If one perturbs the system by employing the uniform transverse magnetic field (and inserts a charged particle) the situation changes. In this case there emerge the KAM tori whereas the phase space is not sharply divided anymore - now it exhibits the hierarchy of KAM-like islands and cantori which are surrounded by the chaotic sea.
3.2 Mushroom billiard - quantum
Quantum mechanically the billiard inside the domain V is described by the time-independent Schr¨odinger equation with the Hamiltonian #(p,q), the eigenenergies Ej and the eigenstates ipj
- ^ipi(q) = Eiipi(q), (3.2)
where h is the Planck constant divided by 2vr whereas V2 is the Laplace operator; we use 2M = h = 1 in the following. In the case of hard walls represented in mushroom billiard the Dirichlet boundary conditions have to be applied: Vn(q)U = 0.
The eigenstates are either mainly regular or mainly chaotic, depending on the phase space region they concentrate on. One clearly sees (Fig. 3.4) the two types of wavefunc-tions which are already classified in (Percival, 1973). In this thesis we have used the desymmetrised mushroom billiard when the foot has been located in the central position, so that we would not have to deal with two symmetry classes of eigenstates.
44
Chapter 3. Dynamical tunneling in mushroom billiards
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
r r
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
r r
Figure 3.3: The radial part NmnJm (jmnr) of the quarter circle eigenfunctions from Eq. 3.3. The upper two plots show these functions for the fixed radial quantum numbers: n = 1 and m = 2,4,. . . ,28 on the upper left plot (black), n = 2 and m = 2,4,.. . ,24 on the upper left plot (red), n = 5 and m = 2,4,. . .,14 on the upper right plot (orange), n = 6 and m = 2, 4,. . .,12 on the upper right plot (violet). The lower two plots represent the fixed azimuthal quantum numbers: m = 2 and n = 1, 2, ... , 8 the lower left and m = 20 and n = 1, 2, . .., 8 the lower right plot.
The regular wavefunctions resemble those of the Hamiltonian H1/4 for the quarter circle. If R = 1, its eigenenergies and eigenfunctions, written in polar coordinates (r,tp) read
L1/4 = i and *P™(r < 1, Y>) = NmnJm (jmnr) sin(mip), ip^(r >1,ip) = 0 (3.3)
These are characterized by the radial (n = 1, 2,...) and the azimuthal (m = 2, 4,...) quantum numbers. Here Jm is the m-th Bessel function whereas jmn indicates its n-th root and Nmn = -t/8/tv/Jm-1(jmn) is the normalisation constant. The radial part NmnJm (jmnr) of the quarter circle eigenfunctions is shown in Fig. 3.3.
In Fig. 3.4 (the right picture) we have resolved the quantum numbers (m, n) for a mushroom regular eigenstate and written down the energies of this regular and the neig-bouring chaotic state; these energies are very close together. As evident from the following dynamical tunneling can, in fact, explain this small splitting.
3.3. Microwave experiments
45
m=26 n=2
1/4
1344.816
Figure 3.4: A neighbouring ’chaotic’ and ’regular’ state in the desymmetrised mushroom billiard. The probability density |?(q)|2 is plotted.
3.3 Microwave experiments
3.3.1 The similarity between QM and ED
By first employing the Maxwell equations one obtains the Helmholtz equation for the propagation of electromagnetic waves (Jackson, 1982)
V2 B + fc2 B
0,
(3.4)
with a wavenumber k = ^, where z/ is the frequency and c is the speed of light. Now one applies the cartesian coordinates and the cylindrical waveguide with walls from any ideal conductor, with any cross section oriented along the z direction. Using the ansatz
E(x,y,z,t) B(x,y,z,t)
Eq. (3.4) can be simplified to
1 f E(x,y)ei(±kzZ~27TVt) = B(x,y)ei(±kzZ~27TUt)
[v
x,y
k
nfEl B
0.
(3.5)
(3.6)
The boundary conditions for the EM field at the ideal conducting boundary have to be taken into account: Etang = 0 and Bnorm = 0 where neither the electric field at the surface along the axis z nor the surface currents are allowed. If one encloses the infinite long waveguide at z = 0 and z = d, one gets the resonator of thickness d.
At this point we will select the components of the field parallel to the axis z (all the other components can be derived from them). The boundary conditions are written as
Ez(x,y,z)\Bcrandary = 0,
dBz(x,y,z) dn
I Boundary
0.
(3.7)
_
_
_
46
Chapter 3. Dynamical tunneling in mushroom billiards
In the z direction the condition kz = *f with n = 0,1,2,... has to be fulfilled as well. There are two classes of solutions: the transversal-electric (TE) and the transversal-magnetic (TM) modes. For both, respectively, one finds
Bz(x,y) = E(x,y)cos(^\ n = 0,1,2,..., Bz(x,y,z) = 0 TM (3.8)
where B(x,y) and E(x,y) are scalar fields.
At lower frequencies u < vG = 2d corresponding to the wavenumbers k<% only TM modes with n = 0 (kz = 0) appear whereas all the other modes are exponentially damped. Eq. (3.6) is reduced to
{?2 ,v + k 2)E(x,y)=0 (3.9)
with the Dirichlet boundary condition 0, and S(v) = 0 for v < 0, where hj is the height, ?j the width and Vj is the resonant frequency including the shift, and j in principle runs over all the resolvable resonances up to Nmax. We have actually only evaluated a window of about NHI = 20 resonances at once and, then, the solutions were glued together. If one decides to apply the HI technique, the spectrum from Eq. (3.13) has to be transformed to the time domain via the Fourier transform
S(t)
1
2^
/ S(u) e~2lli{vi~iVi)tdy = S(t) - ^ h- e-2^(^-iri)* (3.14)
for t > 0, and S(t) = 0 for t < 0, with the amplitude hj and the complex frequency Vj - %?j from the j-th state.
Since the measured data are presented in a discrete form, the second term of expression (3.14) has to be rewritten in a discrete form as well, which results in a system of nonlinear equations
S(kr)
N hi
V^ 7 -2m{vj-iYj)kT
3=1
N hi i=i
0,1,...,2Nmax
1.
(3.15)
1
_
v
_
k
1
1
_
_
_
50
Chapter 3. Dynamical tunneling in mushroom billiards
Since we would like to solve the system for z j = e"2™^-^ and hh we will apply the stable Decimated Linear Predictor method (Main et al, 2000). Normally one obtains a much greater number of solutions than the estimated number of eigenvalues in this window region represented by the Weyl formula (see the next section). If one wants to avoid the spurious solutions, one has to select the maximal suitable width ?max and the minimal suitable height hmin of the resonances. Then, one moves one time step r to the left and recalculates the HI. Thus, one estimates the error and decides which calculated resonances are appropriate and which have to be rejected. The error has to be below a previously estimated value Errmax. One should emphasise, that (i) a cut-off of the modulus square \^ (t)\2 of the Fourier transform has been employed, so that the constant background can be eliminated (Kuhl et al., 2008) and, (ii) the border of the window has not been taken into account due to the problems that could arise. We have used the procedure, which was implemented by R. H¨ohmann (H¨ohmann, 2004). Fig. 3.7 shows the reflection spectrum and its corresponding HI fit which only slightly differ from one another. The advantages of the harmonic inversion are:
• The parameters of the Lorentzians can be determined for many resonances simultaneously even if these strongly overlap with each other, which, for example, happen when the width is 10 times larger than the mean resonance spacing, as recently presented in (Kuhl et al., 2008);
• The procedure is stable, so it can be applied to many data;
• No starting points are needed for this procedure. The determination of the starting points in the case of overlapping resonances is a big disadvantage of the conventional fit procedure. Therefore, HI can be applied in higher frequency region.
There are also certain disadvantages to the harmonic inversion such as:
• There appear certain the deviations if one tries to determine the pole positions in cases where two neighbouring resonances lie very close to one another and their amplitude is very large. Deviations also emerge with small resonances;
• The optimal filter parameters ?max, hmin and Errmax depend on the spectrum, so one has to select such parameters that one eventually obtains as many desired resonances as possible;
• There are some unresolved (about 15%) and a few redundant resonances. Therefore, after the calculation one has to check the entire spectrum and, additionally, fit the missing resonances by applying the conventional fit procedure and eliminate all the redundant ones.
3.4 Level Statistics
As soon as one has calculated the eigenfrequencies Uj of the microwave billiard, one has already obtained the eigenenergies Ej of the corresponding QM system: Ej = ^k) = *§v]. At this point one should emphasise again that in microwave billiards the eigenfrequencies are systematically shifted to the left, i.e. microwave eigenvalues are slightly smaller than the QM ones (Stein et al., 1995).
3.4. Level Statistics
51
Here we will consider the total number of states N(E) of a 2D quantum billiard system below the energy E. This function is termed the spectral staircase function and it fluctuates about its average value (N(E)), which only depends on the area A, on the circumference L, on the curvature and on the edges of the billiard K. This dependence is represented by the generalised Weyl formula
(ME)) = — (AE - lVe] + K. (3.16)
1
If one wants to compare the fluctuations Nflvc = N(E) - (N(E)} in the regions with different (N(E)), one has to unfold the spectrum first, i.e. the spectrum has to be transformed in such way that the average density of states remains the same for all the states treated. e can be defined as e = (N(E)) so that N(e) = e + Nfluc(e), which means that the mean level spacing of N(e) equals 1. The unfolded energy levels e,- are subject to the statistical analysis where the level spacing distribution P(S), which was already discussed in Chap. 2, is chosen as the main statistical measure.
3.4.1 Level dynamics
If one wants to measure level dynamics, one has to change the parameter of the system and observe how its eigenvalues behave during the changes. In the experiment presented the position of the foot b was moved in 1 mm steps from the central to the outermost right position.
The motion of the energy eigenvalues, employed as a function of an external parameter, resembles the dynamics of the particles of a one-dimensional gas with the repulsive interaction (Pechukas, 1983; Yukawa, 1985; St¨ockmann, 1999). Like in particle dynamics there also exists a conservation law for the eigenvalues employed as a function of an external parameter. The lower plot in Fig. 3.8 is obtained from the numerical calculation (described in detail in Sec. 3.5) where almost all the eigenvalues connect into the so-called spaghetti (Barth et al., 1999) appear. If, on the other hand, one measures the eigenfrequencies experimentally with only one antenna, spaghetti are broken due to the missing resonances (upper plot of Fig. 3.8) since the antenna hits the nodal lines of the wavefunctions. In the experiment presented here 37% of the resonances are missing. We believe that the missing resonances are randomly, not systematically, distributed. But there are no missing spaghetti, so one can identify the neighbouring eigenvalues, which is absolutely necessary if one wants to calculate the level spacing distribution. In this way the experiment is fully autonomous and it does not depend on the numerics.
Although the position of the foot of the mushroom billiard constantly changes, the total area and the circumference always remain constant, so all the spectra have the same mean density and the number of levels represented in Eq. (3.16). One can also notice that certain spaghetti are represented as mere straight lines up to a certain parameter value and as ’wavy’ lines as soon as the parameter value is exceeded. Up to that parameter value these levels correspond to the regular levels. All the ’wavy’ spaghetti correspond to the chaotic levels. Some straight-lined spaghetti are long whereas the others are short. This can be explained in terms of the radial quantum number n of the semicircle which counts the number of ’hills’ of the absolute square of the wavefunction in the radial direction. The same quantum numbers can be attributed to the regular wavefunctions as well. Thus, the regular wavefunctions with smaller n (and larger m) ’survive’ longer if one changes
52
Chapter 3. Dynamical tunneling in mushroom billiards
er
o
+
Q
et
o
+
Q
200
150
100 - f*^o**** J^L xtkfm Cr^l«:
&* 15*J*<**, L 1^%I 1 JFll v* 1
50 u Kur **^ l V** fS H *
0
4.54 4.56 4.58 4.60 4.62 4.64 4.66 4.68
Frequency v / Ghz
150 1
100
4.54 4.56 4.58 4.60 4.62 4.64 4.66 4.68
Frequency v / Ghz
Figure 3.8: The microwave mushroom billiard levels from Fig. 3.1 with the foot moved in 1 mm steps. The radius of the semicircle equals R = 400 mm, which is the same as the diameter 2a of the foot where the depth l = 100 mm. There are 200 different positions from the central to the outermost right position. For the sake of better comparison the units in numerical plot are the same as the ones from the experimental plot although these are not real units in numerical calculations. All the details are explained in the text.
3.5. Expanded boundary integral method
53
the position of the foot, which is due to the fact that these functions mainly ’live’ in the vicinity of the border of the semicircle (see Fig. 3.3).
One has to admit that in level dynamics there also appear the misconnections, so one can wrongly determine the neighbouring eigenvalues. But this error is not as frequent as the eigenvalues disappearing from a single spectrum.
3.4.2 The level spacing distribution - comparison with the RMT prediction
Fig. 3.9 compares the mushroom experimental spectra (Fig. 3.8) with the RMT prediction treated in Chap. 2. One notices that the best fitting Gaussian model for PDAnBR is very suitable for the experimental data. Similarly, the best fitting exponential model is of comparable quality. In both cases a appears as the only fitting parameter whereas p is determined by classical dynamics.
For each experimental spectral data we have used 21 configurations by employing equidistant positions of the foot around the central values b + a - R = 10 mm (the upper plot) and 30 mm (the lower plot). For each configuration we have used the spectral stretches in the interval of 100-300 consecutive levels. The parameter p for these configurations varies up to ±7.5% (the upper plot) and ±8.5% (the lower plot) around the central values indicated in the plots. In the experimental spectra the level spacing distribution is affected by an antenna and by tunneling as well although, experimentally, the two effects cannot be separated. For the Fig. 3.9 we have chosen the best fitting plots although the curve may not fit so well in certain cases. One can, thus, conclude that this experiment, in fact, well agrees with our theory although it does not confirm it completely.
3.4.3 Absorber in the foot
As evident from Fig. 3.2 the classical chaotic orbits enter the foot whereas the regular ones remain outside of it. Due to this the regular and the chaotic eigenvalues can be separated by the microwave experiment. This can be achieved if a carbonate absorber is inserted into the bottom of the foot (Fig. 3.10). Thus one creates an open system, which absorbs the energy, so that the chaotic eigenstates disappear in the end. We have qualitatively obtained the expected results, i.e. the chaotic states are mainly absorbed whereas the regular ones remain intact (Fig. 3.11).
3.5 Expanded boundary integral method
We have calculated the numerical eigenvalues from Eq. (3.2) by using the expanded boundary integral method (EBIM), developed in (Veble et al., 2007) and based on the classical boundary integral method (BIM) described in (B¨acker, 2003). With the help of these two references one can describe the basic characteristics of the method. The method is general, so it can be applied to any billiard shape (which is its greatest advantage) although it is rather slow (which is its greatest disadvantage).
Eq. (3.2) can be represented by the k form:
(V2 + fc2)V(q) = 0,
(3.17)
54
Chapter 3. Dynamical tunneling in mushroom billiards
1.0
CO
Co
0
2
1.0
0
2
P =
On.=
0.236 0.139 0.188 1790
Figure 3.9: The comparison between the experimental data (histogram) and the best fitting theoretical curves for Pß%RN: The solid red curve represents the Gaussian model, the dash-dotted black curve indicates the exponential model, the dashed blue curve is the BR distribution (with the same p) whereas the dotted brown curve represents the Wigner distribution. aG and (q) = - / Y0(k\q - q(s)\)u(s)ds. (3.21)
4 Jsv
Now one has to solve Eq. (3.19). There are several possibilities for this; usually one tries to find the zeros of the determinant det(A(k)), although, typically, certain levels, especially those close to degeneracy, disappear in this case. This problem can be solved by expanding the matrix A(fc) in the Taylor series around a chosen reference value k0 (Veble et al., 2007), so that one can determine the solution(s) k close to the reference value: k = k0 + 5k where 5k must be small. From the Eq. (3.19) one obtains
[a(M + 5kA'(k0) + ^^A"(fc0) + ... I u = 0, (3.22)
where A'(k0) and A"(k0) are the first and the second derivative of the matrix A with respect to the k at the point k0 which is obtained by calculating the derivatives of each matrix element. If only the first two terms on the left side are selected, one obtains the generalised eigenvalue problem
A(fc0)u0 = -5koA'(ko)u0, (3.23)
where u0 and 6k0 are the first orders of the expansions of u and 6k respectively. In the case of the mushroom billiard the accuracy of Eq. (3.23) is, numerically, one order of magnitude higher than expected: if k0 varies about the final solution, the eigenvalues
2The convex corners were treated in (Okada et al., 2005) by changing the starting Eq. (3.18).
3.5. Expanded boundary integral method
57
from Eq. (3.23) are the third order polynomial in 5k as opposed to the general second order polynomial that was expected.
Thus, Eq. (3.23), which was selected in the end, is very suitable in this case. After solving it, one has to decide which 5ks are good solutions. Larger 5ks are rejected since they do not agree with the starting assumption of the expansion. Each diagonalisation is characterised by a number of good solutions which also depend on the accuracy desired. In our calculation of the mushroom billiard from the ground state up to k2 = 5000, there appear up to 3 appropriate eigenvalues per diagonalisation. While the accuracy of the method is of the third order in 5k0 (see Eqs. (3.22, 3.23) and the text below Eq.(3.23)) and the mean level spacing is proportional to 1/k0, we have chosen the steps of ?Uko) = 0.05(^L)-1/3, so that similar errors in the units of the mean level spacing
( k 0 Start
appear across the entire region of the calculation. Such a step dependence ?k0(k0) might not be crucial and the constant ?k0 = 0.05 would probably not be so inappropriate, but it was used and it applied well in (Veble et al., 2007) in the case of the Monza billiard, so we decided to use it for the mushroom billiard as well.
In the non-convex geometry the BIM can provide exterior solutions for the Neumann boundary conditions as well although this did not happen when EBIM was employed in the calculation of the mushroom billiard levels. If one wants to check the results roughly, one has to apply the Weyl formula and observe the difference N(E) - {N(E)} which should oscillate around zero. If there are too many or too few eigenvalues, N(E) - {N(E)} has a jump up or down, which immediately indicates that an eigenvalue is redundant (spurious) or that it is missing. In such cases one focuses on the critical region and repeats the calculation. Usually the shift in initial value k0 suffices, so that one manages to obtain a missing level or eliminate a redundant one.
3.5.1 Corners and the accuracy of EBIM
The method and its accuracy have been tested on the 3/4 circle billiard with R = 1. The results of the comparison between the numerical and the exact (analytic) solutions are shown in Fig. 3.12 where the upper right plot corresponds to the equidistant points around the boundary. One can notice that the errors in units of the mean level spacing are not small. Since in our EBIM the boundary points must not be located at the corners, we believe that corners are the most important source of the errors. For all the corner, which are rational fractions p/q of vr any eigenfunction can be expanded into a convergent series of the Fourier-Bessel functions
ip(r, 9) = V aiJqi (Jar) sin (^o] , (3.24)
where r and 0 are the polar coordinates starting at the corner whereas ai are the expansion coefficients. If one expands the Bessel functions J^hr) for small r values (i.e.
p qi
around the corner), one obtains J^hr) oc rp, which indicates the following: the smaller
p
the exponent q, the larger the value of the ip(r,e) for small r values. The 3/4 circle is characterised by two vr/2 corners and by a single 3vr/2 corner. Around the latter the wavefunction are larger which, consequently, influences the results in a larger extent than the vr/2 corners. For this reason the points were condensed around the 3vr/2 corner; first with a 3 times larger constant density from the corner in the interval up to 0.2 R. Due
58
Chapter 3. Dynamical tunneling in mushroom billiards
0.03
0.02
0.01
0.00
ijM
jMw^KK^-j^ImmA
200
400
600
800
1000
0.008 abs(dEunf)
0.006
0.00^
o.oo;
0.000
i
0.005
abs(dEunf)
0.004
0.003
0.002
0.001 0.000
0
200
4-00
600
0
200
400
600
Figure 3.12:
\dEunf\
N(E
numerics j
tained for the 3/4 circle compared with
the t
N(Ejtheory) theoretical
eigenvalue errors numerically ob-(zeros of Bessel functions) for the 1/4
circle (only) in units of the mean level spacing. The quantities in the upper right plot are the same as in the lower two plots. The Ns on the abscissa (alone in this plot) correspond to the consecutive indices js of the eigenvalues of the 3/4 circle. All the details are explained in the text.
to this, the results shown in the lower left plot of Fig. 3.12 are largely improved. If one increases the density of points, the calculation requires more time. Bearing in mind both the accuracy and the CPU time we have decided to employ a denser mesh of points by a factor 1 to 4 which linearly increases over an interval of length 0.4 R away from the corner (the lower right plot from Fig. 3.12); the same procedure was followed in the case of the mushroom calculations as well.
Fig. 3.13 shows the first 9 lowest eigenstates of the mushroom billiard whereas Fig. 3.14 represents a few excited eigenstates with their regular, nonlocalised and localised chaotic (bouncing-ball or scarred) wavefunctions. In the upper plot from Fig. 3.15 one notices the difference between the eigenvalues calculated for B = 12 and those calculated for the three times larger B = 36 (for / = a = §R). Almost all the eigenvalues increase as soon as one applies a larger number of discretisation points per wavelength. The largest absolute difference is 0.0122 whereas the mean absolute difference equals 0.0023. The lower plots from Fig. 3.15 represent the eigenstates with the largest difference; one can see that the 3vr/2 corner is still problematic since these wavefunctions have large amplitudes around it.
_
3.5. Expanded boundary integral method
59
16.603
E 4 = 97.906 E 5 = 104.758
146.786
E o = 149.554
Figure 3.13: The lowest 9 wavefunctions of the mushroom billiard calculated with EBIM (l = a = 11 09R). The regular states are red, the chaotic ones are blue whereas the ground state is black. The probability density with 8 equidistant contours from 0 up to the maximal value is plotted. The upper three plots also show the boundary points.
60
Chapter 3. Dynamical tunneling in mushroom billiards
J239
3054.600
J281
3591.810
Er
290
3696.067
<&
>J»
iT • * ÄS*
ÖO 9SrO Q©0>
J295
3743.215
E302= 3839-612
3901.587
4tVi« ji«
J«M8foS
$ .„cd
'31-
3993.639
J316
4006.63^
125.914
Figure 3.14: The same as Fig. 3.13, but here 9 higher exited states are represented.
3.5. Expanded boundary integral method
61
dEunf 0.002
0
50 100 150 200 250 300
E gi = 1237.890
E168= 2198.556
Epog = 3653.805
* \ ;V \ i©. i
0 *
B36
the difference between the
Figure 3.15: The upper plot: dEunf = N(EjB1 2) - N(Ej
numerically obtained eigenvalues for the mushroom billiard with the configuration l = a = for two different numbers of discretisation points per wavelength, i.e. for B = 12 and for B = The lower plots: the eigenstates with the largest eigenvalue errors.
10
R
19
36.
In the next step all the eigenvalues with the difference ? 0.009 were tested one more time, with the six times bigger B = 72, which resulted in the maximal difference 0.0152. Bearing in mind all this the maximal numerical error of the calculation for B = 12 was estimated to 0.02 of the mean level spacing. One should be aware that this is the largest possible error whereas the mean error is more than 5 times smaller.
3.5.2 Level dynamics - increasing the depth l
In addition to level dynamics described in Subsec. 3.4.1 we have calculated another one, i.e. by deepening the foot of the mushroom billiard. This method is better since the regular eigenstates do not disappear in this case, which is indicated by a full-length
62
Chapter 3. Dynamical tunneling in mushroom billiards
IrX^.v >Lx*Ai
1020
o
CD r—
x 1000
o en
- 980
cni
^T it
' .vX>^ JU
X X * XX X X
- X X xl'x X X X v XX X &-
1-83.5
2484.0
2484.5
E
2485.0
Figure 3.16: The numerically obtained levels for the central (b = R-a) desymmetrised mushroom billiards with the foot depth l increasing in steps of ?l = 1/760 ? 0.00139. The radius of the semicircle is R = 1, the width a = 10/19 ? 0.526. There are 1029 different positions, beginning with l = 0 and continuing equidistantly up to lmax = 257/190 ? 1.35263. The energy E = k2 is presented on the abscissa whereas the parameter, linearly dependent on the depth of the foot l, appears on the ordinate. The lower right plot represents the level dynamics around one regular state (showed in the lower left plot) for all possible parameter variations; the regular state is, in this case, surrounded by a cluster of chaotic levels.
vertical straight lines in the level dynamics plots (Fig. 3.16). This level dynamics was also presented by our second experiment which is described and used in Subsec. 3.7.2, although the experimental eigenvalues have not been connected into the spaghetti.
3.5. Expanded boundary integral method
63
2300 2320 2340 2360 2380 2400 2420 2440
K
E177= 2303.776 E179= 2343.004 E186= 2419-228
Figure 3.17: The upper plot: level dynamics showing the bouncing-ball states as bent, vertical structures. The lower plot: three bouncing-ball states for the parameters l = a = 11 09R.
First, the eigenvalues were calculated for l = lmax in each small interval ?k0. In the next configuration with l = lmax - ?l (where ?l indicates the step size), the eigenvalues from the previous step were used as initial eigenvalues. One can repeat this procedure until l = 0 is reached. Due to the decreased level density the eigenvalues were even shifted appropriately since, on average, the eigenvalues of the next configuration are larger than those from the previous one. In this way we have managed to improve our calculation to such an extent that it became faster by a factor of about 4.
At this point one should briefly mention the so-called bouncing-ball states which also appear in our system (the lower plots of Fig. 3.17). These states are localised along the classical parabolic periodic orbits. Contrary to the regular states their eigenvalues are
64 Chapter 3. Dynamical tunneling in mushroom billiards
represented by bent vertical ’lines’ (structures) in the level dynamics plot (the upper plot in Fig. 3.17), which indicates that they are not as strong as the regular states, but that they are, in fact, stronger in some positions and weaker in the others - this depends on the position of the foot of the mushroom billiard since these eigenstates may also be localised outside the foot. There exists a correlation between the slope of these vertical ’lines’ and the strength of the bouncing-ball modes, i.e the stronger (or purer) a bouncing-ball mode the more vertical the corresponding ’line’ in the level dynamics plot. We have also seen some scarred eigenstates (the upper left plot from Fig. 3.14) which are localised along the classical hyperbolic orbits and whose ’lines’ in the level dynamics plot, as expected, are much more bent than the ’lines’ indicating the bouncing-ball states.
3.5.3 The level spacing distribution - comparison with the RMT prediction
Figs. 3.18 - 3.22 show the results of the comparison between the mushroom numerical spectra (Fig. 3.16) and the tunneling RMT prediction treated in Chap. 2. One notices that the best fitting Gaussian model for P%%R appropriately describes the numerical data. Similarly, the best fitting exponential model is suitable in this case as well. In both cases a is the only fitting parameter whereas p is determined by classical dynamics.
We have employed sequential levels from jmin to jmax for 101 geometrical configurations of the equidistantly varied parameter / in steps of ?/ = 1/760 « 0.00139 around the central position /0. The parameter p for these configurations varies only up to 4% around the central value indicated in each plot.
The lower plot from Fig. 3.22 represents the enlarged interval for small S. Due to the numerical error 0.02 of the mean level spacing (see Subsec. 3.5.1) the histogram in the interval S = [0, 0.04] is not reliable. Figs. 3.18 - 3.22 prove that the numerical histograms well agree with the best fitting theoretical curves describing the tunneling effects from Chap. 2. But in order to confirm the theory even more stronger, one should improve it and one should also obtain more precise numerical results for S => 0.
3.6 Avoided crossings
The narrowly avoided crossings, where ?E indicates the separation at the closest approach (splitting), are especially interesting. Such crossings, for example, appear in the study of atomic systems perturbed by external fields, in the molecular systems employed as a function of the internuclear distance, etc.
Let us consider the level dynamics of the Hamiltonian written as
H = H1 + \H2) (3.25)
with the fixed Hamiltonian H1 and with the parameter A dependent second term \H2. The parameter from a regular system is changed in such a way that the system remains regular, there only appear normal crossings but no avoided crossings appear. Thus, the distribution of the regular crossings can be represented by
PRegnlar(?E) = 5(?E).
(3.26)
3.6. Avoided crossings
65
(/}
to
l.U
p = 0.195
ctg= 0.091
0.8 cte= 0.092
/""'••.. N = 9694
0.6 ^\ —
0.4 /fl ¦' 1-
?/ \ -
0.2 —
0.0 1 1 1 1 1 1 1 1 1 1 1 1 II 1 _iii
0
1.0
2
0.215 0.098 0.099 10096
0.4 ^
0.0 t 0
2
Figure 3.18: The comparison between the numerical data (histogram) for (jmin,jmax) = (300,400) (the upper plot) and (jminjmax) = (200,300) (the lower plot) and the best fitting theoretical curves for P]j%RN: The solid red curve represents the Gaussian model, the dash-dotted black curve is used for the exponential model, the dashed blue curve indicates the BR distribution (with the same p) and the dotted brown curve represents the Wigner distribution. aG and aE are the best fitting values of a for the Gaussian and for the exponential model respectively. N is the number of objects in the histogram. The billiard parameters: a = 10/19, R = 1 (both plots), l0 = 1.287 (the upper plot) and l0 = 1.024 (the lower plot). All the details are explained in the text.
However, the avoided crossings appear if the H is strongly chaotic in the full energy range of varying ?. Here we have considered time reversal invariant systems obeying the
66
Chapter 3. Dynamical tunneling in mushroom billiards
CO
Co
1.0
0.8
0.6
0.4^
0.2-•
0.0 E 0
1.0
0
P
Oc =
0.227 0.100 0.105 10100
2
= 0.240 = 0.108 = 0.123 = 10100
2
Figure 3.19: The same as Fig. 3.18 for (jmin,jmax) = (200,300) (both plots), l0 = 0.892 (the upper plot) and l0 = 0.761 (the lower plot).
GOE universality class, provided that the symmetry of the system does not change if one changes the parameter ? . On could, in fact, obtain a simple analytic formula for the avoided-crossing distribution if one assumed that such crossings are isolated, i.e. that the third level is so far away that one can neglect its influence, which means that each avoided crossing can be treated as a two-level mechanism. One may even make a weaker assumption, i.e. the multiple avoided crossings appear less frequently, so they are statistically insignificant. Then one may locally consider only two strongly interacting levels. Thus, one can select a two-dimensional subspace of two neighbouring levels which meet
3.6. Avoided crossings
67
to
0
1.0
0.8
0.6 -I
0.4
0.2 -¦
0.0 t 0
2
2
P =
(Jr. =
N =
0.255
0.109
0.114
4601
Figure 3.20: The same as Fig. 3.18 for (jmin, jmax) = (100, 200) (the upper plot), (jmin, jmax) (300,400) (the lower plot) and l0 = 0.629 (both plots).
in the avoided crossing. In this subspace the effective Hamiltonian may be represented by the 2 x 2 matrix in the basis where H2 is diagonal (Zakrzewski et al., 1993; St¨ockmann, 1999). Thus, one obtains
H
a b ^ V\ 0
c d +X0 v2
(3.27)
where a, b, c, vl, v2 are real numbers. A simple explicit calculation shows that the dependence of the levels on the parameter A can be represented as a hyperbola whose two
_
_
68
Chapter 3. Dynamical tunneling in mushroom billiards
to
1.0
0.8
0
2
1.0
0.8
0.6
0.4
0.2 -•
0.0
0
2
P = CTg =
CTE =
N =
0.272 0.089 0.105 9897
P 0.291
ar, 0.119
CTF 0.130
N = 9989
Figure 3.21: The same as Fig. 3.18 for (jmin,jmax) = (100,200), l0 = 0.497 (the upper plot) and UminJmax) = (200,300), l0 = 0.366 (the lower plot).
branches E12(\) can be calculated
E1 2(A) = ±-\l((v1-v2)\ + a-b) 2 + Ac2 + - ((V1 + v2)X + a + b). (3.28)
The minimal gap AE between the energy levels E1 and E2 as function of A appears at A0 = -*=*- and equals 2 Id. Since H obeys the GOE universally class, the matrix element c is normally distributed according to RMT. Due to this the probability distribution of
avoided crossings reads P(AE) = ^exp (-fj^) where a is a free parameter. Since
3.6. Avoided crossings
69
co
l.U
p = 0.313
ctg= 0.099
0.8 cte= 0.118
- N = 9898
0.6 •, f*\ ¦-C^- ^\ —
• t \ Lvl V
i,—*
0.4
0.2 - ; -
_.* **. r^*^.
0.0 i i i i 1 1 1 1 1 1 1 1 1 1 1 1 1
0
to
0.00
2
1.0
0.8
0.6 — p
0.4 0.2 ^ flM i-l J^TT
00
0.05
0.10
0.15
0.20
Figure 3.22: The same as Fig. 3.18 for (jmin,jmax) = (100,200), l0 = 0.234 (the upper plot) and enlarged (the lower plot). All the details are explained in the text.
the ratio of the average avoided crossing to the average spacing is a system dependent quantity (Zakrzewski et al., 1993), one has to normalize P(?E) to unit average avoided crossing to obtain a=L=, which results in
22
Pgoe(?E)
7T
exp
?E2
n
(3.29)
Since the crossings are non-generic, the probability of the level crossings in the chaotic systems with one system parameter equals zero. Due to this one has to change two or more system parameters to create a degeneracy. When only one parameter is varied, there appears a finite closest approach ?E to the eigenvalues E1(?) and E2(?) rather
2
_
70
Chapter 3. Dynamical tunneling in mushroom billiards
than a degeneracy. It was numerically confirmed that, for different systems, Eq. (3.29) appropriately describes what is happening in chaotic systems (Zakrzewski et al., 1993).
We would like to deal with the avoided crossings in the mixed-type systems, especially in the mushroom billiards in the case where one deepens the foot (see Subsec. 3.5.2). There exist two types of avoided crossings: the RC avoided crossings, which appear between a regular and a chaotic level, and the CC avoided crossings, which appear between two chaotic levels (Fig. 3.23). The RC avoided crossings appear due to tunneling, which affects the system; their size is employed as a tunneling measure. The two-hyperbola fit (see Fig. 3.23 - the upper and the middle plots) was applied to each (RC or CC) isolated avoided crossing. For practical reasons the two hyperbolas E1,2(\) are described by a somewhat different set of five parameters a0, a1, a2, a3 and a4: E1,2(\) = ±v/a20(\ - a2)2 + a21 + a3 + a4(A- a2) where the new parameters are simple functions of a, b, c, v1, v2 from Eqs. (3.27, 3.28). The minimal energy distance ?E = 2\a1\ occurs at A0 = a2. For the two-hyperbola fit we have used about 20 points whereas in the experimental data fewer points were often available due to the missing resonances.
If the density of states at the local scale does not vary much, one can apply the two-hyperbola fit on the energy or on the Weyl axes - one only switches the avoided-crossing distances to another scale if necessary. For the purpose of this section avoided-crossing distances have to be placed on the Weyl axes. Therefore, we have first unfolded the spectra up to the E = 4200 and, then, we have employed the two-hyperbola fit, so that we eventually devised the histogram of the splittings at the avoided crossings. For the sake of comparison with the theoretical prediction, the histogram has to be normalised to unit area and to unit average splitting at avoided crossings. The lower plot from Fig. 3.23 represents the avoided-crossing distribution of the mushroom billiard levels obtained numerically.
For the CC avoided-crossing distribution the results are in accordance with the theoretical prediction for fully chaotic systems. However, this does not happen in the case of larger crossings which cannot be adequately described by the 2 x 2 model since they cannot be treated as isolated pairs of levels. Another exception is the ratio of the mean avoided crossing to the mean level spacing (?E)/(S). In the Hamiltonian from Eq. (3.25) the mean level spacing explodes to infinity for a big A . If one wants to avoid this explosion, the Hamiltonian has to be expressed by (Zakrzewski et al., 1993)
Jff = cos(A)Jff3 + sin(A)Jff4, (3.30)
where H3 and H4 are GOE random matrices with the same mean level spacing. Then the mean level spacing of H from Eq. (3.30) does not dependent on A any more. With this H the theoretical prediction results in {?E)/{S) = V2 - 1 « 0.41. Simulations with large (N = 50) GOE random matrices yield (?E)/(S) « 0.52 (Zakrzewski et al., 1993). The results differ because the nonisolated avoided crossings are also treated in the GOE model. In real chaotic systems (?E)/(S) is a system dependent quantity, which is usually larger than the 2 x 2 RMT theoretical prediction. This equals 0.51 for the kicked-top model, 0.49 for the Africa billiard and 0.42 for the magnetised hydrogen atom (Zakrzewski et al., 1993). There appear certain discrepancies (i) because the avoided crossings in real systems cannot be regarded as isolated as well as (ii) due to the localised bouncing-ball and scarred eigenstates. If a bouncing-ball or a scarred eigenstate and another chaotic eigenstate, which create an avoided crossing, are dislocated from one another, so they do
3.6. Avoided crossings
71
70.0 70.5 71.0 71.5
K
72.0
1.90 70.00 70.10 70.20 70.30 70.40 K
71.50 71.60 71.70
K
71.80
1.0000
0.0010
0.0001
N(RC) = 6847 N(CC) = 3055
0
a AE
Figure 3.23: The upper plot: the avoided crossings in a numerically calculated level dynamics fitted on the Weyl axis; the RC crossings are indicated by the red colour whereas the CC ones are blue. The dotted lines represent the eigenvalues for the quarter circle. The middle plots: the magnification of a CC (blue) and a RC (red) avoided crossing where the closest distances are indicated by small horizontal coloured straight lines whereas the black straight lines represent the asymptotes of the hyperbolas. The blue number at a regular state indicates the azimuthal quantum number m divided by 2 whereas the red number is the radial quantum number n. The lower plot: the distribution of splittings for each of the cases, where the solid blue curve represents the Eq. (3.29). N(RC) and N(CC) are the numbers of objects included in each histogram. For the sake of comparison the first moment of the RC histogram is normalised with respect to the first moment of the CC histogram. K is defined in Eq. (3.16).
72
Chapter 3. Dynamical tunneling in mushroom billiards
not overlap very strongly, which results in smaller splittings reducing the (?E}/{S). In our system this ratio for the CC avoided crossings equals
^ff»0.27. (3.31)
The value does not change if the quantities are scaled in the appropriate way, i.e. the mean level spacing (S)M for the entire spectrum is scaled with the mean level spacing (S)c for the chaotic states and the mean CC splittings with (S)c as well since both quantities were divided by the same constant. Thus, the final result from (3.31) greatly differs from the theoretical 2x2 RMT prediction value (0.41) and, even more greatly, from the N x N value (0.52). We believe that the discrepancy occurs (i) due to the mixed-type nature of the system; the CC avoided crossings, their gaps and even their existence also depend on the neigbouring regular states; and (ii) due to several bouncing-ball and scarred eigenstates appearing. This could be studied in the future as well. Nevertheless, we are convinced that, due to this result, the validity of Eq. (3.29) could be extended to the mixed-type systems.
In the RC case (red) the average distances are much smaller than those from the CC case, and the ratio {?E)RC/{S}aU of the mean avoided crossing and the mean level spacing of the entire spectrum is about 0.07. We have not managed to describe the RC histogram (for all the regular states) with an analytical function since we do not expect the universal behaviour. But, one can predict the tunneling rates for each regular state separately, which is discussed in the following section.
3.7 Tunneling rates
This section considers the tunneling rates in the mushroom billiard. First one should mention a typical example of tunneling under the barrier, i.e. the 1D double well problem (Landau and Lifshitz, 1989) with the symmetric potential
U(x) = a(x 2 -x 2 0)2 . (3.32)
Quantum mechanically, two quasi-modes with the same energy can be constructed: i.e. one, ipL, in the left well and the other, ij>R, in the right well. The eigenstates are constructed if one employs a symmetric and an antisymmetric linear combination from
4>+ = -17= (4>l + 4>r), 4>- = -17= (4>l - 4>r) (3.33)
with the energies Es and EA respectively. The splitting e = Es-EAin the semiclassical limit is given by
1 +xi
-T - W)\dx
e = -^exp-- \p(x)\dx, (3.34)
where the integral goes over the barrier, ±xx are the two turning points of the trajectories from the right as well as those from the left well respectively, i.e. the points where the moment p vanishes (see Fig. 3.24) whereas u corresponds to the angular frequency of the classical trajectory.
3.7. Tunneling rates
73
x,
Figure 3.24: The double well potential U(x) from Eq. (3.32). If the initial state of the system is ipL, its time evolution U(t)ipL can be indicated by
^Ml = 2 ^ (t)(V- + V>+) = 2e-^/ft(V- + e~^l\h). (3.35)
Thus, its absolute square, the probability \U(t)^L\2, oscillates between \^L\2 and |V>b|2 with the time period t0 = nh/e, provided that the two-level approximation is justified. In the case of small splittings the time period t0 is very long and this is why these modes are called quasi-modes.
Only two close-lying states were treated here. In the case of many (continuum) final close-lying states the qualitative behaviour of the system differs: there appears the exponential decay of the probability of the initial state, so one can apply Fermi’s golden rule here. However, at times larger than the Heisenberg time th = ^, where ?E is the mean level spacing of the discrete spectrum, there can occur quasi- or almost periodic oscillations which depend on the discrete energy specter.
3.7.1 Fermi’s golden rule
Fermi’s golden rule indicates the transition rates in systems with a perturbation. It is derived from the time-dependent perturbation theory although the perturbation appearing in Fermi’s golden rule is time independent. The time-independent Hamiltonian is written as
H = H0 + V, (3.36)
where H0 is the unperturbed Hamiltonian while V indicates the perturbation. If |V>°) and \ip°k) are two different eigenstates of the H0, their matrix elements of the perturbation may be represented in terms of the full Hamiltonian: i.e. (^\V[tpl) = (^\H\ipl). As mentioned earlier the eigenstates \ipj) of the full Hamiltonian are either mainly regular or mainly chaotic. On the basis of these two types we have introduced purely regular IVVeg) and purely chaotic orthogonal states |V>ch> (B¨acker et al., 2008b) which in the first
74
Chapter 3. Dynamical tunneling in mushroom billiards
approximation are the eigenstates of H0. The Appendix B describes how these states can be calculated. One may doubt the existence of the H0. Although H0 cannot be explicitly represented as a slight variation of H, it is well defined by its eigenstates \i$) and their corresponding eigenvalues E]
3
If one wants to obtain the E] eigenvalues, the spectrum of H has to be regarded as a function of / and every RC avoided crossing has to be replaced by an exact crossing whereas the CC avoided crossings remain intact.
The exponential tunneling decay e~^ of a purely (unperturbed) regular state | ^ reg) into the chaotic states | ^ ch) is described by the tunneling rate 7. For the systems with a finite phase space (like ours) this exponential decay mostly occurs up to the Heisenberg time t m'/2, which could probably explain the deviations between the experimental results and the theoretical prediction (see Figs. 3.27 and 3.28).
Tab. 3.1 shows that the higher the energy of a regular state, the more avoided crossings are formed. The fraction of the ’unreliable’ hyperbolas increases with the increasing m at a fixed n. The avoided distances are in many cases small, which can be explained by the fact that many Bessel functions have larger values at a larger radius, so the hyperbolas resemble straight lines.
Thus, this is the very first experimental confirmation of any theoretical prediction of tunneling rates in billiards without the fitting parameter, with the agreement extending over more than 2 orders of magnitude. This is the most important result in our study of the dynamics of the avoided crossings in the mushroom billiard.
3.7.3 Tunneling rates - the numerics
With the numerically obtained levels the tunneling rates have been calculated in the same way as the experimental tunneling rates and their errors; the eigenvalues are the same as the ones employed in the analysis of the level spacing distribution model treated
3.7. Tunneling rates
81
10.0000
1.0000
0.1000
7mn
0.0100
n=6
n=5
0.0010
10.0000
1.0000
n = 6
n = 5
0.0010
0.0001
a=4
Figure 3.29: The same as Figs. 3.27 and 3.28 for numerically calculated levels with EBIM. The upper plot represents all the hyperbolas considered whereas in the lower plot the distance 0 is applied to the ’unreliable’ hyperbolas. The ’numerical resolution’ of an avoided crossing is indicated by the horizontal dashed line.
82
Chapter 3. Dynamical tunneling in mushroom billiards
in this thesis. (see Subsecs. 3.5.2 and 3.5.3). The results are presented in Fig. 3.29 and Tabs. 3.2 and 3.3. Obviously, the numerical results confirm the theoretical findings up to 4 orders of magnitude, which indicates that the agreement in this case is by a 1.5 order of magnitude better than it was in the experiment case.
The dashed horizontal line indicates the error which appeared when we tried to resolve a gap at an individual avoided crossing by our method, which is estimated to be 0.004 in units of the mean level spacing (see Subsec. 3.5.1). This error corresponds to the tunneling rate 7 of about 0.0003 where k changes from 7.5 to 68. In this case all the gaps ?E larger than 0.1 x 2 ?Eerror (see Subsec. 3.7.2) could still be resolved via the two-hyperbola fit of the avoided crossings where ?Eerror is the error of a single level prior to unfolding.
As expected, one notices that the ratio of the fitting and the statistical error is smaller than the one from the experimental study. Due to this and due to a larger number of hyperbolas the standard error is smaller than the one obtained in the experiment. As evident from Tabs 3.2 and 3.3 some (m,n)s contain only the ’unreliable’ hyperbolas, so, consequently, these are not presented in the lower plot from Fig. 3.29. We believe that all these 7mras and, at least, also (m/2,n) = (19,4) should not be considered in our analysis. In an extreme situation one might select the distance zero for the ’unreliable’ hyperbolas, although it would probably be better if one chose the mean of the fitted distance and the distance zero (namely ?E/2), however, such an approach is not necessarily useful.
Numerical accuracy is a large problem here since (as already mentioned above) the corners, especially the non-convex corner of 3vr/2, are the main sources of errors.
After all the eigenvalues had been calculated, there emerged a completely new, a very efficient and highly accurate numerical method, especially adapted for the mushroom billiard (Barnett and Betcke, 2007). This method is based on a set of basis functions &, which are termed particular solutions. These obey the Helmholz equation (3.2) or (3.9) although they do not individually fulfil the Dirichlet boundary condition. For this reason one should obtain such values for Ej, that the specific linear combination of the corresponding basis functions would disappear at the boundary. These linear combinations would, then, be regarded as good approximations for the eigenfunctions whereas the Ejs for the eigenvalues. The authors of the method employed the previously-mentioned Fourier-Bessel functions represented by
& = ,hi (y/Er) sin (—O) , (3.47)
as a set of basis functions. The non-convex corner is selected as the center of the coordinate system. These functions accurately capture the behaviour at the corner.
Since the group from Dresden had already calculated the energy spectra using the new method, we have not performed our own calculation with this method. Their calculation and the results for the tunneling rates are briefly represented in Appendix C. The results agree with the theory at over 18 orders of magnitude, which indicates an extremely strong confirmation.
Even if we tried to improve our EBIM method by treating the influence of the corners more precisely, the agreement would probably not be improved for more than a few orders of magnitude. We have solved the problem by using a general numerical method which is, consequently, slower and less accurate. Nevertheless, we can conclude that the theoretical prediction has also been confirmed by the numerical results.
3.7. Tunneling rates 83
m/2 n NhypALL NhypREL NhypREL NhvpALL if5<260<2a0,
Ä arcsin f, if 260 < 5 < 2a0, 2 ^ ^
s
j^o arcsin ^ - arecos ^ , if 2a0 < S < 2-N/a§ + 6§, 0, if S > 2^ag + 6g.
If one selects 60 > a0 instead of 60 < a0, one should only interchange a0 and 60 in the above formulae.
4.1.2 Exponential distribution
The distribution functions of matrix elements are defined as
9.(a) = ±e-™, *(6) = V*H. (4.5)
For the level spacing distribution function one obtains
P(S) = ^sf dip exp(--(Xa cos ip+ Xb sin ip)\, (4.6)
which cannot be analytically evaluated in a closed form. For small values of S the linear level repulsion law is recovered, i.e.
P(S) « ^^ S. (4.7)
For large values of 5 we use Aacos^ + A6sin^ = isin(^ + (p), with 0 = arctan(Aa/A6), and Ä = y/\l + XI Then the integration variable 0 results in 2vr, so that, at small values of S, P(S) « S/(2na0b0), which well agrees with Eq. (2.14). The integral in (4.11) can be calculated analytically, resulting in
P(S) = A____<*y^ + PW+*
2ira0b0 (a2 + ß2 + a2 ß^VTTa2\JY^ß2
where a2 = S2/(4a2) and ß2 = S2/(46g). The asymptotic behaviour of P(S) at large 5 is characterised as an inverse quadratic power law,
P(S) « 4(a; + M (4.13)
implying that, due to divergence, P(S) does not have the first moment.
4.1.4 Singular times exponential distribution
The normalized singular distributions are defined as
9a(a) = Ca\a\-^e-XM, gh{b) = Cb\b\-^e-XbW, (4.14)
with the normalization constants
Q = A|"w/(2r(l -fa)). (4.15)
Here % = a, b and the exponents m < 1 whereas Y{x) is the gamma function. These distribution functions are singular but integrable power laws for a, b -> 0 and they decay nearly exponentially in the tails. For the level spacing distribution it follows
ns) = ca s (s)^+H) f '^H^+AtSiM), (4,6)
but this formulae still have no analytical solutions. One can evaluate it for small argument S where the exponential can be approximated by 1. This results in the following level repulsion law, i.e. the fractional exponent power law
/ q -(ßa+ßb) TVl _
2~ 2T(1
P{S) = CaCb S- 2 %/2 2\ (4.17)
The power law distribution of matrix elements leads to the power law repulsion in the level spacing distribution.
4.1.5 Comparison with numerics
Fig. 4.1 shows the numerically evaluated P(S) for 2D random matrices defined by the four ensembles mentioned above and compared to the theoretical curves. An excellent agreement is observed for certain typical parameter values.
Chapter 4. Numerical studies of non-Gaussian real symmetric random 88 matrices
Figure 4.1: The numerical histograms are compared with the theoretical curves for the four ensembles from Sec. 4.1 (not unfolded, except for the singular times exponential case). M is the number of matrices taken from the ensemble. Top-left: Box distribution, a0 = \/2/2, b0 = 1/2, M = 108. Top-right: Exponential distribution, Aa = y/2, Xb = 1, M = 108. Bottom-left: Cauchy-Lorentz distribution a0 = 1/2, b0 = 1/2, M = 108. Bottom-right: Singular times exponential distribution: Aa = y/2, Xb = 1, /x„ = 0, /x6 = 1/2, M = 107; this distribution is unfolded.
4.2 Numerical calculations with higher dimensional non-Gaussian random matrices
We have generalized the random matrix ensembles from Sec. 4.1 from N = 2 to the higher dimension N by applying the distribution ga(a) for all the diagonal elements of the N-dimensional real symmetric matrix and the distribution gb(b) for all the off-diagonal matrix elements.
This study primarily intends to determine whether there exists the transition (in the cases 4.1.1 and 4.1.2), or maybe there is not any (in the cases 4.1.3 and 4.1.4), from the nonuniversal behaviour at small N values to the Hackenbroich-Weidenmu¨ller (HW) universal GOE behaviour at N = ?. In the numerical studies of the spectra we have used two different types of spectral unfolding which is necessarily approximate due to the finite dimension of the spectra. For small matrices we have calculated the level spacings for the M representatives from the given ensemble (M is the number of matrices drawn from the ensemble). The spacings obtained have then been divided by the mean spacing calculated for the entire set. In the case of large matrices this rule does not make sense since the density of eigenvalues is strongly non-uniform. Therefore, in the case of larger matrices, we have applied the phenomenological rule (see Subsec. 2.5.1), which means that we have selected a certain number of the nearest neighbours (typically unf = 20 for
4.2. Numerical calculations with higher dimensional non-Gaussian random matrices 89
N = 120), so that the local mean level spacing has been calculated by averaging the entire set. One should emphasise that N = 120 and unf = 20 in the phenomenological unfolding procedure means 120 - 20 = 100 effective energy levels (for comparison: in Sec. 2.5 only the effective points have been treated). The two unfolding procedures differ in the mean used; this mean can be either global or local, so it is sometimes hard to decide which type of mean is to be applied in the calculation. The results (unavoidably) depend on the unfolding method selected although, in most cases, its influence is not as large as one might expect at first sight.
4.2.1 Box (uniform) distribution
As observed in Fig. 4.2 the transition from N = 2 to GOE is quite fast since for N = 3, 4 one still notices certain deviations from the GOE (approximated by the Wigner distribution) whereas at N = 7 the agreement with the Wigner distribution is already perfect despite the fact that we have used the unfolding procedure for small matrices in this case (the procedure is described in the previous section). At the dimension N = 120 one observes perfect agreement with the GOE distribution for both the unfolding procedures, i.e. for small matrices as well as for the 20 neighbours-rule. This demonstrates that, at least in this case, the final results relatively weakly depend on the unfolding procedure. The last plot in Fig. 4.2 shows the eigenvalue distribution for the dimension N = 120, so one clearly sees that the limiting distribution is smooth and confined to a finite interval. Since the assumptions from the HW theorem are fulfilled in this case, the transition to the GOE behaviour is expected (see the plots from Fig. 4.2).
4.2.2 Exponential distribution
In comparison with the previous subsection the transformation of the level spacing distribution as a function of N is relatively slower here, but it completes at N = 120, which confirms the theoretical prediction (as observed in Fig. 4.3). The level spacing distribution approaches the Wigner distribution already at N = 4. Then the value of N increases from 7 to 10 and 20 so that the agreement is almost perfect. The last plot in Fig. 4.3 re-confirms this for the dimension N = 120. Like in the case of the box distribution the distribution of the eigenvalues for N = 120 is smooth and confined to a finite interval (but this is not shown in the figure).
4.2.3 Cauchy-Lorentz distribution
The Cauchy-Lorentz distribution is interesting, since its first moment diverges. One also knows that according to Eq. (4.13) the tail of P(S) forN = 2 is proportional to 1/S2, so it does not have the first moment either. In the limit of very large Ns, such as N = 120, one clearly sees (Fig. 4.4) that the density of the eigenvalues is not confined to a finite interval. Thus, one of the assumptions from the HW theory is not fulfilled in this case, so in the limit N -> oo one might expect certain deviations from the universal GOE behaviour. This is indeed what one notices in Fig. 4.4, which shows the level spacing distribution for N = 120 for the unfolding procedure with unf = 20 neighbours. At smaller Ns, such as N = 3, 4, 7 and, also, at N = 120 if one uses the unfolding procedure typical of small matrices, we have observed an unusual behaviour of P(S) which differs
Chapter 4. Numerical studies of non-Gaussian real symmetric random 90 matrices
Figure 4.2: The numerical histograms compared with the Wigner curve for the ensemble with the box distribution for a0 = ?2/2, b0 = 1/2. M is the number of matrices from the ensemble. Top-left: N = 3, M = 106, unfolding for small matrices. Top-right: N = 4, M = 106, unfolding for small matrices. Middle-left: N = 7, M = 106, unfolding for small matrices. Middle-right: N = 120, M = 104, unfolding for small matrices. Bottom-left: N = 120, M = 104, unfolding with unf = 20 neighbours. Here the curve represents the exact GOE result. Bottom-right: the eigenvalue distribution for N = 120, M = 104, showing that the distribution is smooth and confined to a finite interval.
in each case; clearly, this behaviour is influenced by the fact that the mean energy level spacing approaches infinity when N › ?. If certain singularities of the Cauchy-Lorentz distribution of the matrix elements are eliminated (by cutting off the tails at a large but finite a, b), one observes the immediate transition to the GOE behaviour, which well agrees with the HW theory (see Fig. 4.4). The cutting is realised via the transformation from the uniform distributed random numbers abox, bbox to the Cauchy-Lorentz distributed random numbers a, b. This transformation is represented by
4.2. Numerical calculations with higher dimensional non-Gaussian random matrices 91
Figure 4.3: The numerical histograms compared with the Wigner curve for the ensemble with the exponential distribution for Aa = y/2, Xb = 1. M is the number of matrices from the ensemble. Top-left: N = 3, M = 106, unfolding for small matrices. Top-right: N = 4, M = 106, unfolding for small matrices. Middle-left: N = 7, M = 106, unfolding for small matrices. Middle-right: N = 10, M = 105, unfolding for small matrices. Bottom-left: N = 20, M = 105, unfolding with unf = 6 neighbours. Bottom-right: N = 120, M = 103, unfolding with unf = 20 neighbours. Here the curve represents the exact GOE result.
a = a0 tan
k abox 2 cl0
b = b0 tan
2 b0
(4.18)
e),a0(l - e)] and
where abox, bbox are uniformly distributed on the intervals [-a0(l - t;,a0v^ -[-60(l-e), 60(l-e)] respectively. Thus, a, b are Cauchy-Lorentz distributed with ga(a), gb(b)
from Subsec. 4.1.3 up to the border acut = a0 tan f (1 - e) and bcut = b0 tan f (1 - e) where \a\ < acut and |6| < bcut. The value e = 0.01 has been chosen for the last plot from Fig. 4.4.
Chapter 4. Numerical studies of non-Gaussian real symmetric random 92 matrices
Figure 4.4: The numerical data for the ensemble with the Cauchy-Lorentz distribution for a0 = V2/2, 60 = 1/2. M indicates the number of matrices selected from the ensemble. For the sake of comparison the thin curve represents the Wigner distribution. Top-left: the density of the eigenvalues for N = 120, M = 103 which is obviously not confined to a finite interval. Top-right: N = 120, M = 103, unfolding with unf = 20 neighbours, the smooth curve is the Wigner distribution for the sake of comparison. Middle-left: N = 3, M = 106, unfolding for small matrices. Middle-right: N = 4, M = 106, unfolding for small matrices. Bottom-left: N = 7, M = 106, unfolding for small matrices. Bottom-right: N = 120, M = 103, unfolding with unf = 20 neighbours. Here the tails of the matrix element distribution function have been cut off at the specific value of e = 0.01, due to which the level spacing distribution immediately approaches the GOE behaviour.
4.2.4 Singular times exponential distribution
The singular (times exponential) distribution from Sec. 4.1.4 (4.14) at the dimension N = 2 might trigger off a completely new phenomenon, i.e. the fractional power law level repulsion (see Fig. 4.1). Surprisingly, if this ensemble is generalised to the higher
4.2. Numerical calculations with higher dimensional non-Gaussian random matrices 93
Figure 4.5: The numerical histograms compared with the Wigner curve (except for N = 120 where the exact GOE appears) for the ensemble with the singular times exponential distribution for A„ = y/2, \b = 1, ßa = 0, and ßb = 1/2, except in the last plot (bottom-right) where [ib = 0.9. M indicates the number of matrices chosen from the ensemble. Top-left: N = 3, M = 105, unfolding for small matrices. Top-right: N = 4, M = 105, unfolding for small matrices. Middle-left: N = 7, M = 105, unfolding for small matrices. Middle-right: N = 120, M = 5 x 103, unfolding with unf = 20 neighbours. Here the curve represents the exact GOE result. Bottom-left: N = 120, M = 5 x 103, the eigenvalue density. Bottom-right: N = 3, M = 105, unfolding for small matrices.
dimensions of N > 2, for pa = 0 and pb = 1/2, one observes a transition to the linear level repulsion already at the dimension N = 3. Fig. 4.5 shows a relatively fast transition to the GOE distribution appearing with the increasing values of N. One can verify that the level density for N = 120 is confined to a finite interval, so the HW prediction applies in this case. If one increases the value of /ib to /ib = 0.9, i.e. closer to the nonintegrable singularity /ib = 1, this influences the P(S) behaviour so strongly that the fractional power law level repulsion can be observed in the end.
Chapter 4. Numerical studies of non-Gaussian real symmetric random 94 matrices
Figure 4.6: The numerical histograms compared with the Wigner curve for the ensemble with the singular times exponential distribution for Aa = y/2, Xb = 1, ßa = 0, and ßb varying from 0.9 to 0.9999. In all the cases the dimension of the matrices was N = 120 and the unfolding was calculated via unf = 20 neighbours. M indicates the number of matrices chosen from the ensemble. Top-left: ßb = 0.9, N = 120, M = 2 x 103. Top-right: ßb = 0.95, N = 120, M = 2 x 103. Middle-left: ßb = 0.97, N = 120, M = 2 x 103. Middle-right: fib = 0.99, N = 120, M = 2 x 103. Bottom-left: ßb = 0.999, N = 120, M = 103, one can also observe the Poissonian curve P(S) = exp(-S). Bottom-right: fib = 0.9999, N = 120, M = 2 x 104 and the Poissonian curve P(S) = exp(-S) as well.
However, as the exponent /ib increases and approaches the nonintegrable singularity ßb = 1, in the case of large matrices with N = 120 there appears the transition from the GOE (at smaller values of ßh) to the Poissonian (exponential) if ßb is sufficiently close to the value 1 (see Fig. 4.6). At the intermediate values of ßb, such as ßb « 0.99, we have conjectured the fractional power law level repulsion, althoug additional analytical as well as numerical analysis is needed to provide more quantitative predictions and descriptions.
95
Chapter 5
Summary and conclusions
This thesis mainly discusses dynamical tunneling (a concept introduced in (Davis and Heller, 1981)) in the systems with the mixed (i.e. regular-chaotic) type dynamics. We have considered two different aspects: (i) we have studied the level spacing distribution for a general mixed-type system and tested the theory for the mushroom billiard levels and (ii) we have analysed the tunneling rates as well as the avoided-crossing distribution for the mushroom billiard. Additionally, we have calculated the level spacing distribution for the random matrices of non-Gaussian ensembles in order to model the fractional power-law level repulsion.
In Chap. 2 we have devised a new random matrix model which describes the distorted Berry-Robnik level spacing distribution affected by the tunneling between the regular and the chaotic states, denoted by PEbrn(s). We have derived a two-level analytic formula for PEbr(S) which agrees with the results for higher dimensional random matrices at not too large values of the coupling parameter. We have also developed an analytic two-level model creating the level spacing distribution P$BR(S) for the all-to-all level couplings, which well applies in the cases of general perturbation such as the presence of an antenna in a microwave resonator. Since this all-to-all two dimensional model, where all the levels are coupled, perfectly agrees with the iV-dimensional simulations denoted by Pdbrn(s), this is, certainly, a very successful model. Its exact integral representation PßBR(S) can be analytically expressed in a closed form for small as well as for large values of S. The same applies for the tunneling-distorted Berry-Robnik distribution PlBR(S) where only the regular and the chaotic levels are coupled. However, an overall good, simple analytic approximation has not been devised yet. We have also shown that the non-Gaussian models for the all-to-all couplings (i.e. the exponential and the box distributions for the off-diagonal matrix elements) well agree with the Gaussian model although they deviate from it significantly a the intermediate values of S. However, they almost perfectly agree with the corresponding two-dimensional model.
The experimental and the numerical data for the mushroom billiard (see Chap. 3) can be well described by the theoretical models employed. If the energy increases towards infinity (the semiclassical limit), a should approach zero (and there appears the BR behaviour), which does not necessarily happen quickly and monotonously. We have noticed that a somehow decreased (but not monotonously), but this cannot be confirmed by the results obtained, so one cannot make any definite conclusions at this point. The reason, presumably, lies mainly in the theory since two important things were not taken into account, i.e (i) the splittings of regular states due to the presence of chaotic states and
96
Chapter 5. Summary and conclusions
(ii) localisation effects. There may be other reasons as well, i.e. the experimental and the numerical inaccuracy of the mushroom levels or, even, the specific properties of the very mushroom billiard, such as scarred (Dietz et al., 2007) or bouncing-ball eigenfunctions. In the future we intend to improve the theory in the way discussed in Sec. 2.6, so that it can be tested in other systems, billiards or maps, or even in the recalculated mushroom billiard levels with the new method of particular solutions. In the case of the mushroom billiard the semiclassical limit is reached very quickly, i.e. already after a few thousands of levels. For the numerical level dynamics in this high lying regime one should definitely apply the method of particular solutions since the EBIM would be too slow in this case. Experimentally, without using the superconducting cavity - in (Dietz et al., 2007) the authors resolved the first 938 resonances and some additional ones as well, but they did not reach more resonances since they only measured one billiard configuration at different antenna positions whereas we reached the 600th branch in the level dynamics measurement - obviously, this semiclassical limit is unreachable.
According to the overall picture of the level spacings in the mixed-type systems (Robnik and Prosen, 1997), there exists the regime of linear level repulsion at small S whereas the regime of the fractional power law level repulsion appears at larger S followed by the Berry-Robnik tail. Thus, the theoretical approach presented seems to be very promising. The linear level repulsion, which always exists due to tunneling, is a very robust phenomenon indeed (Grossmann and Robnik, 2007b).
Recently, V.A. Podolskiy and E.E. Narimanov have tried to correct the Berry-Robnik level spacing distribution by modelling the tunneling effects in another way, which has resulted in the formula presented in (Podolskiy and Narimanov, 2003b) and based on (Podolskiy and Narimanov, 2003a). However, this derivation is incomplete and inconsistent since it does not incorporate the normalisation and since the physical grounds and the results are questionable, so this approach is not discussed in any further details.
Localisation effects should cause the fractional power law level repulsion, as captured by the Brody-like distribution (Prosen and Robnik, 1994a,b). We have tried to model localisation in Chap. 4 of the thesis where we have numerically studied four non-Gaussian ensembles of real symmetric random matrices for various dimensions N, from N = 2 to N = 120, defined by (a) the box, (b) the exponential, (c) the Cauchy-Lorentz and (d) the singular times exponential distribution. We have studied the transition to the universality regime at N -> oo, precisely described by the GOE from the random matrix theory, as predicted in (Hackenbroich and Weidenmu¨ller, 1995), provided that two conditions are fulfilled, i.e. (i) the limiting eigenvalue distribution is smooth, and (ii) it is confined to a finite interval.
In the first two cases (a) and (b) the two conditions are fulfilled, so the HW prediction is confirmed whereas in the case (c) the level spacing distribution clearly deviates from the GOE curve since the eigenvalue distribution is not confined to a finite interval. At the intermediate values of N the behaviour of P(S) is quite unusual indeed. In the case (d), for the singular times exponential distribution of the off-diagonal matrix elements, only a partial deviation from the HW theorem and the two conditions presented can be observed. This happens at those values of /ib which closely approach to the nonintegrable singularity /ib = 1. Thus, surprisingly, the fractional power law level repulsion, which is clearly manifested in the two-dimensional theory (Grossmann and Robnik, 2007b) and in its simulation, does not appear at the higher dimensions N > 2, except, possibly, for the case of a specific interval of /ib approaching the nonintegrable value 1, e.g. at /ib « 0.99.
97
However, only additional numerical and analytical studies could explain the behaviour of singular random matrix ensembles in more details. Indeed, the Hamilton operators of nearly integrable systems (slightly perturbed integrable systems in the KAM scenario) in the basis of the integrable part are quantally represented as sparsed matrices, which should exhibit singular distribution of matrix elements (Prosen and Robnik, 1993a).
Chap. 3 experimentally and numerically studies the tunneling rates 7mra in the mushroom billiards. Fermi’s golden rule is applied to all possible eigenstates V™ of the quarter circle which are also termed ’pure regular states of the mushroom billiard’ and which decay into the neighbouring chaotic states. The results have confirmed the theory especially devised for the mushroom billiard. We have observed a very good agreement without any free parameters including the error bars, which is unprecedented for billiards. On average an eigenstate of the quarter circle with quantum numbers (m, n) decays by about one order of magnitude slower than the one with quantum numbers (m,n+ 1). This result could be, potentially, applied in the field of microlasers, optical fibres, etc.
Surprisingly, Fermi’s golden rule which is derived for the continuum distribution of levels, well applies in closed systems, i.e. in systems with the discrete distribution of levels. Strictly speaking, this rule should be applied only in the case of the open mushroom billiard where the depth of the mushroom foot is infinite, but it is, obviously, suitable in this case as well.
In theory, the approach using a fictitious integrable system could, hopefully, be applied to other generic billiards where it is much more challenging and difficult to determine an appropriate Hreg. In the experiment presented the unavoidable coupling to the environment was preferred for small tunneling rates; however Fig. 3.27 shows that in the microwave experiment the coupling by the antenna is actually negligible beyond three orders of magnitude. This is a promising aspect for future experimental studies of more complex systems, especially in the cases where the numerical and theoretical results are not available yet.
Similarly, the tunneling rates could also be studied in the full chaotic systems, such as the Sinai or the stadium billiard. Here, one could study the decay rates of the bouncing-ball states into the (pure) eigenstates. In this way, one could learn a lot about the strength of interaction between the bouncing-ball modes and the eigenstates, one could, for example, determine how localisation influences the splittings at avoided crossings, to determine the relationship between tunneling and localisation. Similarly, this could also be studied in the mushroom billiard.
Chap. 3 also presents the results of the numerical study of the avoided-crossing distribution in the mushroom billiard. Both types of splittings, i.e. splittings at CC avoided crossings and splittings at RC avoided crossings, were studied separately. In the case of CC splittings we have found very good agrement with the 2 x 2 RMT prediction. However, one still cannot explain a ratio of the mean splitting and the mean level spacing which is smaller than predicted. Due to these new results, the RMT prediction may eventually be generalised to the CC splittings from the mixed-type systems.
98 Chapter5.Summary and conclusions
BIBLIOGRAPHY
99
Bibliography
Abramowitz, M. and Stegun, I. A. (1970). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, ninth edition.
Abul-Magd, A. Y. and Simbel, M. H. (2000). High-order level-spacing distributions for mixed systems. Phys. Rev. E, 62:4792–4798.
Altmann, E. G., Motter, A. E., and Kantz, H. (2005). Stickiness in mushroom billiards. Chaos, 15(3):033105, 1–7.
Altmann, E. G., Motter, A. E., and Kantz, H. (2006). Stickiness in Hamiltonian systems: From sharply divided to hierarchical phase space. Phys. Rev. E, 73:026207, 1–10.
Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics. Springer, New York, second edition.
B¨acker, A. (2003). Numerical aspects of eigenvalue and eigenfunction computations for chaotic quantum systems. Lecture Notes in Physics, 618:91–144.
B¨acker, A., Ketzmerick, R., L¨ock, S., Robnik, M., Vidmar, G., H¨ohmann, R., Kuhl, U., and St¨ockmann, H.-J. (2008a). Dynamical tunneling in mushroom billiards. Phys. Rev. Lett., 100:174103.
B¨acker, A., Ketzmerick, R., L¨ock, S., and Schilling, L. (2008b). Regular-to-chaotic tunneling rates using a fictitious integrable system. Phys. Rev. Lett., 100:104101.
B¨acker, A., Ketzmerick, R., and Monastra, A. G. (2005). Flooding of chaotic eigenstates into regular phase space islands. Phys. Rev. Lett., 94:054102.
B¨acker, A., Ketzmerick, R., and Monastra, A. G. (2007). Universality in the flooding of regular islands by chaotic states. Phys. Rev. E, 75:066204, 1–11.
B¨acker, A. and Schubert, R. (2002). Amplitude distribution of eigenfunctions in mixed systems. J. Phys. A: Math. Gen., 35:527–538.
Barnett, A. H. and Betcke, T. (2007). Quantum mushroom billiards. Chaos, 17(4):043125.
Barth, M., Kuhl, U., and St¨ockmann, H.-J. (1999). Global versus local level dynamics: the limits of universality. Phys. Rev. Lett., 82:2026.
Berry, M. V. (1977). Regular and irregular semiclassical wavefunctions. J. Phys. A: Math. Gen., 10:2083–2091.
100
BIBLIOGRAPHY
Berry, M. V. (1985). Semiclassical theory of spectral rigidity. Proc. Roy. Soc. Lond. A, 400:229–251.
Berry, M. V. (2002). Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature. J. Phys. A: Math. Gen., 35:3025–3038.
Berry, M. V. and Robnik, M. (1984). Semiclassical level spacings when regular and chaotic orbits coexist. J. Phys. A: Math. Gen., 17:2413–2421.
Berry, M. V. and Robnik, M. (1986). Statistics of energy levels without time-reversal symmetry: Aharonov-Bohm chaotic billiards. J. Phys. A: Math. Gen., 19:649–668.
Berry, M. V. and Tabor, M. (1977). Level clustering in the regular spectrum. Proc. Roy. Soc. Lond. A, 356:375–394.
Betcke, T. and Trefethen, L. N. (2005). Reviving the method of particular solutions. SIAM Review, 47:469–491.
Bohigas, O., Boos´e, D., Egydio de Carvalho, R., and Marvulle, V. (1993a). Quantum tunneling and chaotic dynamics. Nucl. Phys. A, A560:197–210.
Bohigas, O., Giannoni, M. J., and Schmit, C. (1984). Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett., 52 (1):1.
Bohigas, O., Tomsovic, S., and Ullmo, D. (1990). Dynamical quasidegeneracies and separation of regular and irregular levels. Phys. Rev. Lett., 64(13):1479.
Bohigas, O., Tomsovic, S., and Ullmo, D. (1993b). Manifestations of classical phase space structures in quantum mechanics. Phys. Rep., 223(2):43–133.
Brodier, O., Schlagheck, P., and Ullmo, D. (2001). Resonance-assisted tunneling in near-integrable systems. Phys. Rev. Lett., 87:064101.
Brodier, O., Schlagheck, P., and Ullmo, D. (2002). Resonance-assisted tunneling. Ann. Phys., 300:88–136.
Bunimovich, L. A. (1974). On ergodic properties of certain billiards. Funct. Anal. Appl., 8(3):254–255.
Bunimovich, L. A. (2001). Mushrooms and other billiards with divided phase space. Chaos, 11 (4):802–808.
Casati, G., Guarneri, I., and Valz-Gris, F. (1980). On the connection between quantization of nonintegrable systems and statistical theory of spectra. Lette. Nuovo Cimento, 28:279.
Davis, M. J. and Heller, E. J. (1981). Quantum dynamical tunneling in bound states. J. Chem. Phys., 75:246.
Dembowski, C., Gr¨af, H.-D., Heine, A., Hofferbert, R., Rehfeld, H., and Richter, A. (2000). First experimental evidence for chaos-assisted tunneling in a microwave annular billiard. Phys. Rev. Lett., 84(5):867.
BIBLIOGRAPHY
101
Dietz, B., Friedrich, T., Miski-Oglu, M., Richter, A., and Sch¨afer, F. (2007). Spectral properties of Bunimovich mushroom billiards. Phys. Rev. E, 75:035203(R), 1–4.
Dietz, B., Friedrich, T., Miski-Oglu, M., Richter, A., Seligman, T. H., and Zapfe, K. (2006). Nonperiodic echoes from mushroom billiard hats. Phys. Rev. E, 74:056207, 1–8.
Doron, E. and Frischat, S. D. (1995). Semiclassical description of tunneling in mixed systems: case of the annular billiard. Phys. Rev. Lett., 75(20):3661.
Eltschka, C. and Schlagheck, P. (2005). Resonance- and chaos-assisted tunneling in mixed regular-chaotic systems. Phys. Rev. Lett., 94:014101.
Feist, J., B¨acker, A., Ketzmerick, R., Rotter, S., Huckestein, B., and Burgd¨orfer, J. (2006). Nanowires with surface disorder: giant localization lengths and quantum-to-classical crossover. Phys. Rev. Lett., 97:116804.
Frischat, S. D. and Doron, E. (1998). Dynamical tunneling in mixed systems. Phys. Rev. E, 57(2):1421–1443.
Gr¨af, H.-D., Harney, H. L., Lengeler, H., Lewenkopf, C. H., Rangacharyulu, C., Richter, A., Schardt, P., and Weidenmu¨ler, H. A. (1992). Distribution of eigenmodes in a superconducting stadium billiard with chaotic dynamics. Phys. Rev. Lett., 69(9):1296.
Grossmann, S. and Robnik, M. (2007a). On level spacing distributions for 2D non-normal Gaussian random matrices. J. Phys. A: Math. Theor., 40:409–421.
Grossmann, S. and Robnik, M. (2007b). Some generic properties of level spacing distributions of 2D real random matrices. Zeitschrift f¨ur Naturforschung A, 62:471–482.
Guhr, T., Mu¨ller-Gr¨oling, A., and Weidenmu¨ller, H. A. (1998). Random matrix theories in quantum physics: Common concepts. Phys. Rep., 299:189–425, cond-mat/9707301.
Haake, F. (2001). Quantum signatures of chaos. Springer, Heidelberg, New York, second edition.
ˇ Haake, F., Ku´s, M., Seba, P., St¨ockmann, H.-J., and Stoffregen, U. (1996). Microwave
billiards with broken time reversal invariance. J. Phys. A: Math. Gen., 29:5745–5757.
Hackenbroich, G. and Weidenmu¨ller, H. (1995). Universality of random-matrix results for non-Gaussian ensembles. Phys. Rev. Lett., 74:4118.
Hanson, J., Ott, E., and Antonsen, T. (1984). Influence of finite wavelength on the quantum kicked rotator in the semiclassical regime. Phys. Rev. A, 29:819–825.
Hensinger, W., H¨affner, H., Browaeys, A., Heckenberg, N., Helmerson, K., McKenzie, C., Milburn, G., Phillips, W., Rolston, S., Rubinsztein-Dunlop, H., and Upcroft, B. (2001). Dynamical tunneling of ultracold atoms. Nature, 412:52.
Heusler, S., Mu¨ller, S., Braun, P., and Haake, F. (2004). Universal spectral form factor for chaotic dynamics. J. Phys.A: Math. Gen., 37:L31–L37.
102
BIBLIOGRAPHY
H¨ohmann, R. (2004). Untersuchung von Billiards mit symmetrischer Unordnung. University of Marburg. BS thesis.
Ishikawa, A., Tanaka, A., and Shudo, A. (2007). Quantum suppression of chaotic tunneling. J. Phys. A: Math. Theor., 40:F1–F9.
Izrailev, F. M. (1989). Intermediate statistics of the quasi-energy spectrum and quantum localisation of classical chaos. J. Phys. A: Math. Gen., 22:865–878.
Izrailev, F. M. (1990). Simple models of quantum chaos: spectrum and eigenfunctions. Phys. Rep., 196:299.
Jackson, J. D. (1982). Klassische Elektrodynamik. Gruyter, Berlin, second edition.
Jacquod, P. and Amiet, J.-P. (1995). Evidence for the validity of the Berry-Robnik surmise in a periodically pulsed spin system. J. Phys. A: Math. Gen., 28(17):4799–4811.
Kaufmann, Z., Korm´anyos, A., Cserti, J., and Lambert, C. J. (2006). Quantized invariant tori in Andreev billiards of mixed phase space. Phys. Rev. B, 73:214526, 1–4.
Ketzmerick, R., Hufnagel, L., Steinbach, F., and Weiss, M. (2000). New class of eigenstates in generic Hamiltonian systems. Phys. Rev. Lett., 85(6):1214.
Kuhl, U., H¨ohmann, R., Main, J., and St¨ockmann, H.-J. (2008). Resonance widths in open microwave cavities studied by harmonic inversion. Phys. Rev. Lett., 100:254101.
Landau, L. D. and Lifshitz, E. M. (1989). Quantum mechanics (Non-relativistic Theory). Butterworth-Heinemann, Oxford, third edition.
Lee, K.-C. (1989). What makes chaos border sticky? Physica D, 35:186–202.
Lehman, R. S. (1959). Developments at an analytic corner of solutions of elliptic partial differential equations. J. Math. Mech., 8:727–760.
Leyvraz, F. and Ullmo, D. (1996). The level splitting distribution in chaos-assisted tunneling. J. Phys. A: Math. Gen., 29:2529–2551.
Lin, W. A. and Ballentine, L. E. (1990). Quantum tunneling and chaos in a driven anharmonic oscillator. Phys. Rev. Lett., 65(42):2927.
Lin, W. A. and Ballentine, L. E. (1992). Quantum tunneling and regular and irregular quantum dynamics of a driven double-well oscillator. Phys. Rev. A, 45(6):3637–3645.
Main, J., Dando, P. A., Belki´c, D., and Taylor, H. S. (2000). Decimation and harmonic inversion of periodic orbit signals. J. Phys. A: Math. Gen., 33:1247–1263.
Malovrh, J. and Prosen, T. (2002). Spectral statistics of a system with sharply divided phase space. J. Phys. A: Math. Gen., 35:2483–2490.
Markarian, R. (1993). New ergodic billiards: exact results. Nonlinearity, 6:819–841.
Mehta, M. L. (1991). Random Matrices. Academic Press, New York, San Diego, second, revised and enlarged edition.
BIBLIOGRAPHY
103
Meyer, H.-D. (1985). Theory of the Liapunov exponents of Hamiltonian systems and a numerical study on the transition from regular to irregular classical motion. J. Chem. Phys., 84(6):3147–3160.
Mouchet, A., Miniatura, C., Kaiser, R., Gr´emaud, B., and D., D. (2001). Chaos-assisted tunneling with cold atoms. Phys. Rev. E, 64:016221, 1–15.
Mu¨ller, S., Heusler, S., Braun, P., Haake, F., and Altland, A. (2004). Semiclassical foundation of universality in quantum chaos. Phys. Rev. Lett., 93:014103.
Narimanov, E. and Stone, A. (1999). Quantum chaos in quantum wells. Physica, 131:221– 246.
Okada, Y., Shudo, A., Tasaki, S., and Harayama, T. (2005). On the boundary element method for billiards with corners. J. Phys. A: Math. Gen., 38:6675–6688.
Pechukas, P. (1983). Distribution of energy eigenvalues in the irregular spectrum. Phys. Rev. Lett., 51:943.
Percival, I. C. (1973). Regular and irregular spectra. J. Phys. B: Atom. Mol., 6(9):L229– L232.
Pineda, C. and Prosen, T. (2007). Universal and nonuniversal level statistics in a chaotic quantum spin chain. Phys. Rev. E, 76:061127.
Podolskiy, V. A. and Narimanov, E. E. (2003a). Semiclassical description of chaos-assisted tunneling. Phys. Rev. Lett., 91(26):263601.
Podolskiy, V. A. and Narimanov, E. E. (2003b). Universal level spacing distribution in quantum systems. arXiv:nlin/0310034v1 [nlin:CD], pages 1–4.
Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. (1986). Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge, etc.
Prosen, T. (1995). Quantum surface-of-section method: demonstration of semiclassical Berry-Robnik energy level-spacing distribution in a generic two-dimensional Hamilto-nian system. J. Phys. A: Math. Gen., 28:L349–L355.
Prosen, T. (1998a). Berry-Robnik level statistics in a smooth billiard systems. J. Phys. A: Math. Gen., 31:7023–7029.
Prosen, T. (1998b). A new class of completely integrable quantum spin chains. J. Phys. A: Math. Gen., 31:L397–L403.
Prosen, T. and Robnik, M. (1993a). Energy level statistics and localization in sparsed banded matrix ensemble. J. Phys. A: Math. Gen., 26:1105–1114.
Prosen, T. and Robnik, M. (1993b). Energy level statistics in the transition region between integrability and chaos. J. Phys. A: Math. Gen., 26:2371–2387.
Prosen, T. and Robnik, M. (1994a). Numerical demonstration of the Berry-Robnik level spacing distribution. J. Phys. A: Math. Gen., 27(13):L459–L466.
104
BIBLIOGRAPHY
Prosen, T. and Robnik, M. (1994b). Semiclassical energy level statistics in the transition region between integrability and chaos: transition from Brody-like to Berry-Robnik behaviour. J. Phys. A: Math. Gen., 27(24):8059–8077.
Prosen, T. and Robnik, M. (1999). Intermediate E(k,L) statistics in the regime of mixed classical dynamics. J. Phys. A: Math. Gen., 32:1863–1873.
Robnik, M. (1983). Classical dynamics of a family of billiards with analytic boundaries. J. Phys. A: Math. Gen., 16:3971–3986.
Robnik, M. (1986). Antiunitary symmetries and energy level statistics. Lecture Notes in Physics, 263:120–130.
Robnik, M. and Berry, M. V. (1986). False time-reversal violation and energy level statistics: the role of anti-unitary symmetries. J. Phys. A: Math. Gen., 19:669–682.
Robnik, M., David, H. M., Vidmar, G., and Romanovskiy, V. (2007). Numerical studies of non-Gaussian real symmetric matrices: confirmation of the Hackenbroich-Weidenmu¨ller theory. Preprint at CAMTP.
Robnik, M. and Prosen, T. (1997). Comment on energy level statistics in the mixed regime. J. Phys. A: Math. Gen., 30:8787–8793.
Robnik, M. and Veble, G. (1998). On spectral statistics of classically integrable systems. J. Phys. A: Math. Gen., 31:4669–4704.
Sheinman, M., Fishman, S., Guarneri, I., and Rebuzzini, L. (2006). Decay of quantum accelerator modes. Phys. Rev. A, 73:052110, 1–12.
Shudo, A. and Ikeda, K. S. (1995). Complex classical trajectories and chaotic tunneling. Phys. Rev. Lett., 74:682.
Sieber, M. and Richter, K. (2001). Correlations between periodic orbits and their role in spectral statistics. Physica Scripta, T90:128–133.
Sinai, Y. G. (1970). Dynamical systems with elastic reflections; ergodic properties of dispersing billiards. Russian Math. Surveys, 25:137.
Steck, D., Oskay, W., and Raizen, M. (2001). Observation of chaos-assisted tunneling between islands of stability. Science, 293:274.
Stein, J., St¨ockmann, H.-J., and Stoffregen, U. (1995). Microwave studies of billiard Green functions and propagators. Phys. Rev. Lett., 75(1):53.
St¨ockmann, H.-J. (1999). Quantum Chaos: An Introduction. Cambridge University Press.
St¨ockmann, H.-J. (2007). Private comunication.
St¨ockmann, H.-J. and Stein, J. (1990). Quantum chaos in billiards studied by microwave absorption. Phys. Rev. Lett., 64(19):2215.
Tanaka, A. and Shudo, A. (2006). Recurrence time distribution in mushroom billiards with parabolic hat. Phys. Rev. E, 74:036211, 1–5.
BIBLIOGRAPHY
105
Tomsovic, S. and Ullmo, D. (1994). Chaos-assisted tunneling. Phys. Rev. E, 50(1):145– 162.
Veble, G., Kuhl, U., Robnik, M., St¨ockmann, H.-J., Liu, J., and Barth, M. (2000). Experimental study of generic billiards with microwave resonators. Prog. of Theor. Ph. Suppl. Num., 139:283–300.
Veble, G., Prosen, T., and Robnik, M. (2007). Generalized scaling method and chaotic time-reversal doubles in quantum billiards. New Journal of Physics, 9:15, 1–17.
Vidmar, G., St¨ockmann, H.-J., Robnik, M., Kuhl, U., H¨ohmann, R., and Grossmann, S. (2007). Beyond the Berry-Robnik regime: a random matrix study of tunneling effects. J. Phys. A: Math. Theor., 40:13883–13905.
ˇ
Seba, P., Haake, F., Kus, M., Barth, M., Kuhl, U., and St¨ockmann, H.-J. (1997). Distribution of the wave function inside chaotic partially open systems. Phys. Rev. E, 56:2680–2686.
Wall, M. R. and Neuhauser, D. (1995). Extraction, through filter-diagonalisation, of general quantum eigenvalues or classical normal mode frequencies from a small number of residues or a short time segment of a signal. I. theory and application to a quantum-dynamics model. J. Chem. Phys., 102:8011.
Wiersig, J. and Hentschel, M. (2008). Combining directional light output and ultralow loss in deformed microdisks. Phys. Rev. Lett., 100:033901.
Wilkinson, M. (1986). Tunneling between tori in phase space. Physica, 21D:341–354.
Wilkinson, M. (1987). Narrowly avoided crossings. J. Phys. A: Math. Gen., 20(3):635– 645.
Yukawa, T. (1985). New approach to the statistical properties of energy levels. Phys. Rev. Lett., 54:1883.
Zakrzewski, J., Delande, D., and Buchleitner, A. (1998). Ionization via chaos assisted tunneling. Phys. Rev. E, 57 (2):1458–1474.
Zakrzewski, J., Delande, D., and Ku´s, M. (1993). Parametric motion of energy levels in quantum chaotic systems. II. avoided-crossing distribution. Phys. Rev. E, 47 (3):1665– 1676.
106 BIBLIOGRAPHY
107
Appendix A
pi calculation for the mushroom billiard
The group from Dresden has analytically derived the classical effective chaotic area of the desymmetrised mushroom billiard for the central position of the foot: b = R - a. The derivation is presented in this appendix together with our numerical results.
The fraction p2 of the chaotic part of the classical phase space volume of the 2D billiard system can be generally represented as
= ?PSC = JdqdpXch(q,p)6(E-H(q,p)) P2 ?ps Jdqdp5(E-H(q,p)) ' ( . )
where ?PS is the volume of the whole classical phase space and ?PSC is the volume of the chaotic part of the classical phase space; Xch(q, p) equals 1 if the trajectory with the coordinate q and the momentum p is chaotic, but if this trajectory is regular, Xch(q, p) equals 0. Due to the relation H(q, p) = 2j = E in billiards the ?PS is written as
f dqdpstE- 2-
?ps = dqdpS E-^—. (A.2)
The integration over the space coordinates provides the area A of the billiard. By replacing the double integral over the momenta with the integral over the energy one gets
?pS = 2nMA f dE5 (e - 2-\ . (A.3)
The remaining integral equals 1, which results in
?ps = 2ttMA. (A.4)
The phase space of the mushroom billiard (Fig. A.1) is given by {(r,^,i?) : (r, a is characterised by regular and chaotic dynamics. Its contribution to Ach, called / should be calculated as
where
Ach
Ia+ 4W
+ 1,
(A.6)
/
1
2^
/f R
I dip I rdr I
0 a
2lT
d&xch(r,(f,&).
(A.7)
Only the radial coordinate r determines whether there appear chaotic trajectory in the hat for r > a (see Fig. A.1). If one considers the trajectories crossing the point (r, reg> and |V>ch> are the eigenstates of different operators, Hieg and Hch, they are not necessarily orthogonal, (V'chlVw) = X with 0 < |x| < 1. In order to apply Fermi’s golden rule the orthonormalised states ^ were introduced |^reg) = |Vw), \M = (\ipch) ~ xlVw))/^1 - Ixl2, leading to
(V'chlVw) = 0.
For the coupling matrix element v = ftM Wreg) = $ch\H$mg) it follows
V = (Am\H - HreglÄeg) + (Äh |#reg|Vw)
1 (V'ch|Jff-Jffreg|V'reg)-^^(V'reg|Jff-Jffreg|V'reg)+0. (B.1)
v71 - Ixl2 v/1-lxl2
The leading order in X for billiards is represented by
v= [ 4>*ch(x,y)(H - Hreg)ijmg(x,y)dxdy + O(x). (B.2)
in
Eqs. (3.38) and (B.2) can be used if one wants to determine the dynamical tunneling
rates in billiards. Here one should emphasise that Hreg and |^g) have to be selected very
carefully, which is a difficult task in the case of a general billiard.
Appendix B. Theoretical analysis of the tunneling rates in the mushroom 112 billiard
This approach can now be applied to the desymmetrised mushroom billiard composed of a quarter circle and a rectangular foot (see Fig. B.1a) where we have chosen R = 1 in the following analysis. For the regular system HYeg the quarter-circle billiard H1/4 is a natural choice (with its eigenvalues E™ and the eigenstates V™ represented in Eq. (3.3)).
(a)
(b)
R
a
Figure B.1: The desymmetrised mushroom billiard: (a) Schematic picture of the coordinate systems used in the theoretical derivation. (b) The auxiliary billiard H1
w
/4
After evaluating Eq. (B.2) one obtains the undefined product of H - Hl/4 = -oo and ipff(x,y) = 0, for y < 0. Therefore, the auxiliary billiard H^4 is introduced (see Fig. B.1b) with a large but finite potential V(x,y < 0) = W > E, characterised by the eigenstates Vi/4W Eq. (B.2) is evaluated in the limit W -»• oo where H^4 approaches #i/4, which leads to
lim
W›?
pa pO
dx dyi)cb(x,y)(-W)i)^w(x,y) Jo-i
dx lim Wip7%w
JQ W^oo ' '
a dtbmn
dx lim l/4,W
-i
(x,y = 0) dyipch(x,y)exp(W
dy
(x,y
o
a dtbmn
dx^(x,y
a dtbmn
. d'r^f
0)-F
VF
V
T?mn ^1/4
y)
w-Lmra
0) dyVch(^,y) lim
Z W->oo
dyVch(x,y)exp( FFexp(FF
(x,y = 0)
(x,y
-i
0)4>ch(x,y
-Nmn
m
dx-Jm(jmnx)^ch(x,y o x
dyi)cb(x,y)2 5(y)
0) 0),
r
v
rprnn ^1/4
FF
y)
^1/4
v)
(B.3)
l
'f
__
a
__
__
__
o
__
__
__
__
__
a
__
113
where the y-integration was performed on ipT%w(x,y) = ip„Zw(x,y = 0)exp(W - E™y),
/a w ( i y) = r i /4 w ( i y = ) exp( yv — j-ja i a obtained from the Schr¨odinger equation for y < 0 and the continuity at y = 0. Further-
chbmn I---------------------
more^(x,y = 0) = JW - E™tl)™w(x,y = 0) has been used, obtained from the
continuity of the derivative at y = 0 and lim^^ V/Wexp( W - E™y) was replaced
by the Dirac delta function; Nmn is the normalisation defined in Sec. 3.2.
One should observe that the value of the chaotic eigenstates |V>ch> is only needed on the line y = 0. For these eigenstates a random wave decomposition (RWD) (Berry, 1977) was employed. Recently, the RWD has been generalised to the systems with the mixed phase space (B¨acker and Schubert, 2002). While this description accurately describes the behavior inside the billiard, it does not incorporate the effect of the boundary, e.g. near the corner, which, for m > 2, largely influences the (final) integral from Eq. (B.3). Therefore, the RWD model (Berry, 2002) was extended to the case of the corner with the angle 3vr/2 using the eigenstates with the Dirichlet boundary conditions (Lehman, 1959),
^(p, tf) ~ Jj J2 Cgj2i W^p) sin (—A' (B.4)
V ch s=l
where the polar coordinates (p, 14 in bi dobili skoraj enak rezultat. Tudi odvisnost rezultatov od velikosti matrike N je zelo majhna.
Rezultati za splošno deformacijo so predstavljeni na sliki i za tunelsko pa na sliki ii. Histogram predstavlja rezultat N dimenzionalne simulacije, medtem ko rdeča krivulja predstavlja 2-dimenzionalni model. Vidimo, daje v primeru splošne deformacije ujemanje za majhne a2 in vse p izredno dobro. Očitno pa je, da 2D model deluje samo dokler so izvendiagonalni elementi majhni v primerjavi s povprečnim razmikom med nivoji. To vidimo na sliki i spodaj desno, kjer predstavimo primer z večjo sklopitvijo o = 0.3.
V primeru tunelske deformacije je ujemanje zadovoljivo, opogumljajoče, a so odstopanja očitna predvsem okrog S = 0. Medtem ko imamo pri N dimenzionalnem modelu linearno odbijanje nivojev, pri 2D modelu odbijanja ni. To opažanje odraža dejstvo, da se v 2D modelu naključne degeneracije v regularnem delu pojavljajo generično, medtem ko so v N dimenzionalnem modelu eliminirane. V slednjem sicer ni direktnih tunelskih matričnih elementov, obstaja pa efektivno tuneliranje drugega reda, ki indirektno sklaplja dve regularni stanji z enim ali več kaotičnimi stanji. Zaradi tega 2D model tukaj odpove.
Imamo dva predloga odprave defekta 2D modela. Pri prvem porazdelitev izvendiagonalnih elementov (iv) zamenjamo z
gb{b) = 2p{l-p)^=ei^-^ +[l-2p(l-p)]—^= exp -^ , (xi)
kjer ax > 0 fenomenološko upošteva razcep drugega reda v regularnem delu spektra. Pri drugem vsak trojček dveh sosednjih regularnih in enega najbližjega kaotičnega nivoja obravnavamo s 3 x 3 matriko, pri čemer je sklopljen samo kaotični nivo z vsakim od obeh regularnih (Stöckmann, 2007). Oboje je še v delu.
Pogledali smo si tudi, kaj dobimo, če je porazdelitev izvendiagonalnih elementov škatlasta ali eksponentna. Izkaže se, da se rezultat spremeni samo nekoliko, večjih spre-memnb pa ni. Kot smo že intuitivno pričakovali, so statistične lastnosti spektrov odvisne predvsem od variance in ne toliko od drugih detajlov porazdelitvene funkcije izvendiagonalnih elementov dane matrike.
ii. Simulacije z naključnimi matrikami
121
1.0
1.0
0.0
0.4 L
Slika i: Rezultati za splošno deformirano porazdelitev BR po razmikih med sosednjimi nivoji Pdbr(s) (rdeča krivulja) in PDA BnRN{S) (histogram) za različne parametre a in p. Vsi nivoji so sklopljeni. Za primerjavo črtkasta modra črta predstavlja porazdelitev BR. Pri obeh krivuljah smo pri izračunu BR uporabili eksaktno GOE porazdelitev. Pri povečavah prikažemo obnašanje histograma in krivulj za majhne S. N dimenzionalni (numerični) in 2D (analitični) rezultati so v primeru a = 0.01, 0.05 in 0.1 praktično identični, krivulja BR pa se razlikuje znatno. Pri velikih a = 0.3 pa vidimo razliko med modeloma. Tam je lepo vidno tudi odstopanje modelov od BR za velike S, ki nastopi zaradi skrčitve oziroma raztegnitve modelskih funkcij, kot smo razložili v tekstu. Pri majhnih S ta efekt ni viden. V primeru s a = 0.01 imamo 10-krat več objektov v histogramu.
122
Daljši slovenski povzetek
s
1.U
n r \f\ ,0 P = 0.75 = 0.05
- jsfjf^**"
0.6 — \ 0.6 Y^ - —
0.4 \ ..2 - -
0.2 n n ......... .........1 .........i.. -
Slika ii: Rezultati za tunelsko deformirano porazdelitev BR po razmikih med sosednjimi nivoji pdbr(s) (rdeča krivulja) in pEbrn(s) (histogram) za različne parametre a in p. Sklopljeni so samo regularni in kaotični nivoji. Za primerjavo črtkasta modra črta predstavlja porazdelitev BR. Pri povečavah prikažemo obnašanje histograma in krivulj za majhne S. N dimenzionalni (numerični) in 2D (analitični) rezultati se relativno ujemajo zadovoljivo, razlikujejo se predvsem okrog S = 0. Na srednji desni in spodnji levi sliki imamo 10-krat več objektov v histogramu.
iii. Gobasti biljard
123
iii Gobasti biljard
Gobasti biljard (Bunimovich, 2001) je sestavljen iz polkroˇzne glave in pravokotne noge (slika iii).
2a
Slika iii: Geometrija gobastega biljarda: R = radij polkroga, 2a = širina noge, l = višina noge, b = pozicija noge (daljša razdalja do roba).
1.0
0.5
0.0
0.5
1.0
0
Slika iv: Zgoraj levo: primer regularne orbite z 70 odboji. Zgoraj desno: primer kaotiˇcne orbite z 100 odboji. Spodaj: fazni portret. Abscisa je koordinata roba s pri odboju, medtem ko je ordinata p sinus odbojnega kota.
Posebna lastnost gobastega biljarda je ostra loˇcenost faznega prostora v natanko eno kaotiˇcno in eno regularno komponento (slika iv spodaj), zato je ta biljard posebej privlaˇcen
124
Daljši slovenski povzetek
za analize. Orbite, ki pridejo enkrat v nogo, so kaotične, vse ostale so regularne (slika iv zgoraj). Regularne orbite se nahajajo v polkrožnem kolobarju nad polkrožno kavstiko z najmanjšim dovoljeni radijem 2a + b - R. Ta kavstika ostro loči regularne orbite od kaotičnih.
Regularni del klasičnega faznega prostora je zelo preprost. Zato je volumen regularnega (ali kaotičnega) dela faznega prostora enostavneje izračunati kot pri KAM sistemih. V skupini R. Ketzmericka v Dresdenu jim je uspelo dobiti analitičen izraz za efektivno kaotično površino gobastega biljarda s centralno pozicijo noge (b = R - a)
Ach = 2la+ \r2 arcsin - + a
Vi?2 - 0?\
(xii)
kjer za delež regularne komponente faznega prostora sledi
A - Ach \kR2 - \ [R2 arcsin (f) + aVR2 - a2]
analitic 0i ——
A
a _________R
la + \kR?
(xiii)
Ta rezultat smo potrdili z numeričnim izračunom z relativno natančnostjo 0.0002.
Medtem ko je regularni del faznega prostora enostaven, pa je kaotični del bolj kompleksen. Ta poleg odbijajočih stanj (ang. bouncing balls) vsebuje tudi druge družine marginalno nestabilnih periodičnih orbit (Altmann et al., 2005, 2006), t. i. ’lepljive’ orbite.
1342.475
1344.812
m=26 n=2
1344.816
Slika v: Verjetnostna gostota sosednjih kaotičnega in regularnega stanja.
Kvantnomehansko biljard opišemo s časovno neodvisno Schrödingerjevo enačbo s hamil-tonko H(p,q), lastnimi energijami Ej in lastnimi stanji ^- V našem primeru imamo Dirichletove robne pogoje: Vn(q)U = 0. Lastna stanja lahko razvrstimo kot v glavnem regularna ali v glavnem kaotična (slika v). Regularne valovne funkcije in pripadajoče energije so podobne lastnim fukcijam in energijam polkroga, kjer pri R = 1 v polarnim koordinatah (r, ) = NmnJm (jmnr) sm{rmp), ^{r >l, 0, izboljšanje teorije in seveda tudi izračun lastnih energij kakšnega drugega sistema mešanega tipa.
vi Izognjena kriˇzanja in tuneliranje
Veje lastnih stanj se lahko križajo ali pa tudi ne, v tem primeru dobimo t. i. izognjena križanja dveh vej z najbližjo razdaljo AE. V regularnem sistemu izognjenih križanj ni, v popolnoma kaotičnem sistemu pa so vsa križanja izognjena. Če ima tak sistem simetrijo na obrat časa, je v njem porazdelitev razcepov v izognjenih križanjih Gaussova, kar sledi iz 2 x 2 GOE modela naključnih matrik za izolirano izognjeno križanje (Zakrzewski et al., 1993). Normalizirana na enotski povprečni razcep se porazdelitev glasi:
128
Daljši slovenski povzetek
1.0
1/3
Co
0
1.0
0.8
0
2
S!
P
Or_ =
0.255 0.117 0.136 9898
2
Slika ix: Primerjava eksperimentalnih podatkov (histogram) z najbolj prilegajočo se teoretično krivuljo P^brn (zgornja) in numeričnih z najbolj prilegajočo se teoretično krivuljo P^brn (spodnja slika), kjer je rdeča krivulja za Gaussov, pikčasto črtkasta črna za eksponentni model, črtkasta modra za BR porazdelitev z istim p in pikčasta rjava za Wignerjevo porazdelitev. aG in (te sta vrednosti najboljših fltov za a za Gaussov in eksponentni model. N je število objektov v vsakem histogramu. Na zgornji sliki imamo nivoje na intervalu (jmin,jmax) = (100,300), na spodnji pa na intervalu (jmin,jmax) = (100,200).
Pgoe(AE)
7T
exp
AL2
n
(xv)
2
_
vi. Izognjena križanja in tuneliranje
129
70.0 70.5 71.0 71.5 K
72.0
170 L 168
CD L^
x 166 o 164
05
^ 162 160 158
C\2
^ x
X X
X X
69.90 70.00 70.10 70.20 70.30 70.40 K
71.50 71.60 71.70 K
71.80
1.0000 E
N(RC) = 6847 N(CC) = 3055 _=
0.0010 -
0.0001
0
1
2
d äE
6
Slika x: Zgornja slika: izognjena križanja v numeričnem izračunu dinamike lastnih energij do k2 = 4200 na Weylovi osi; RC hiperbole so označene rdeče, CC pa modro. Črtkaste črte označujejo lastne vrednosti četrtkroga. Srednji sliki: povečava enega CC in enega RC izogn-jenega križanja; najbližje razdalje so označene s horizontalno barvno črtico, črne ravne črte pa so asimptote hiperbol. Pri regularnem stanju je modra številka azimutalno kvantno število m deljeno z 2, rdeča pa radialno kvantno število n. Spodnja slika: porazdelitev razcepov za vsak tip izognjenih križanj, kjer polna modra črta predstavlja izraz (xv). N(RC) in N(CC) sta števili objektov v posameznem histogramu. Zaradi primerjave je prvi moment RC histograma normaliziran glede na prvi moment CC histograma. K je definiram v formuli (3.16).
130
Daljši slovenski povzetek
Veji lastnih energij Ex in E2 v odvisnosti od zunanjega parametra sistema A imata v bližini izognjenega križanja obliko hiperbol
E1>2(\) = ±a2(X - a2)2 + aj + a3 + a4(X - a2), (xvi)
parametriziranih z a0, al} a2, a3 in a4. Najmanjša razdalja med njima AE = 2 |ai| je pri
Ao = Cl2-
Nas zanimajo izognjena križanja pri gobastem biljardu, do katerih pride med podaljševanjem globine noge /. Tu obstajata dva tipa izognjenih križanj: CC izognjena križanja so med sosednjima kaotičnima stanjema, RC pa med regularnim in sosednjim kaotičnim stanjem. RC izognjena križanja so neposredna posledica tuneliranja v sistemu, njihova najmanjša razdalja pa je mera za tuneliranje. Vsako izognjeno križanje, v primeru numeričnih podatkov smo uporabili okrog 10 točk za vsako vejo, smo po razgrnitvi spektra ’pofitali’ z dvema hiperbolama iz enačbe (xvi), glej slika x.
Izvrednotili smo izognjena križanja do k2 = 4200, izračunali histogram razcepov in ga normalizirali na enotski razcep. Tako se za CC izognjena križanja histogram dobro ujema z 2 x 2 teoretično napovedjo za popolnoma kaotične sisteme, kar nakazuje možno razširitev veljavnosti enačbe (xv) na sisteme mešanega tipa. Neskladje je pri velikih razcepih, ki jih seveda ne moremo popisati z izoliranimi izognjenimi križanji, ter pri kvocientu povprečnega razcepa in povprečne razdalje med sosednjima stanjema, ki je precej manjši od tistega, ki ga napoveduje 2x2 model.
Pri RC-jih je razcep v povprečju veliko manjši kot pri CC-jih, pri njihovi porazdelitvi pa ne pričakujemo univerzalnosti. So pa RC izognjena križanja idelna za kvantitativno analizo dinamičnega tuneliranja. Če bi obravnavali eno samo izognjeno križanje danega regularnega stanja, kar pomeni ’srečanje’ tega regularnega stanja s kaotičnim, ki ima podobno energijo, bi dobili oscilacije verjetnostne gostote iz enega stanja v drugo in nazaj. Tukaj bomo obravnavali koeficiente tuneliranja iz določenega regularnega stanja, upoštevajoč dinamiko lastnih stanj in posledična srečanja tega regularnega stanja s sosednjo vejo kaotičnih stanj. V tem primeru uporabimo Fermijevo zlato pravilo.
Fermijevo zlato pravilo
Fermijevo zlato pravilo pove velikost prehoda med stanji v sistemu s perturbacijo. Tak je tudi naš sistem, le da njegove hamiltonke ne moremo zapisati kot vsote nezmotenega dela in motnje. Eksponentni tunelski razpad e"7* čistega (nezmotenega) regularnega stanja v kaotična stanja, ki traja do Heisenbergovega časa tH, opišemo s koeficientom tuneliranja
7 = 27r(M>ch, (xvii)
kjer je (M2) povprečni matrični element med tem regularnim in kaotičnimi stanji s podobno energijo, pch pa gostota kaotičnih stanj v okolici regularnega stanja. Za gobasti biljard so v skupini v Dresdenu (R. Ketzmerick, A. Backer in S. Lock) izračunali teoretične koeficiente tuneliranja lmn za vsako čisto regularno stanje gobastega biljarda iz enačbe (xiv) in dobili
8 ~, Jm+2s(jmna)2
7™=^ jmUJmn)2 ¦ (xvlll)
vii. Numerična študija negausovskih realno simetričnih matrik
131
Vsota po s izključuje večkratnike od 3 in konvergira zelo hitro. lmn so predstavljeni na sliki xi.
Ker je razcep v izognjenem križanju direktno preko AE = 2\v\ povezan z matričnim elementom v iz (xvii) in smemo pri ne preveliki kaotični komponenti faznega prostora povprečenje razširiti čez pch oziroma Ach iz (xii), dobimo
1=([AE]2Ach/8). (xix)
Koeficienti tuneliranja - eksperiment in numerika
Pri poglabljanju noge smo poleg numeričnega izračuna naredili tudi eksperiment. Za vsako regularno stanje pri eksperimentu s kR med 11.9 in 53.8 in numeriki s k med 11.9 in 67 smo obravnavali vsa izognjena križanja in dobili njihove razcepe v enotah energije pred razgrnitvijo. Z uporabo formule xviii smo dobili koeficiente tuneliranja pri eksperimentalni in numerični obravnavi, rezultati so prikazani na sliki xi. Zaradi velikega deleža nezanesljivih hiperbol pri fitanju, ki so posledica eksperimentalne oziroma nu-merične ločljivosti in prisotnosti antene pri eksperimentu, točke (m/2,n) = (5,1), (7,1), (8,1), (9,1), (11,1), (12,1), (13,1), (11,2), (12,2), (13,2) in (16,4) izločimo iz končne obravnave. Preostale točke se s teorijo ujemajo zelo dobro, zato lahko rečemo, da je to prva eksperimentalna potrditev teorije koeficientov tuneliranja brez dodatnega parametra z ujemanjem, ki je večje od dveh redov velikosti.
Pri numeriki iz istih razlogov (razen prisotnosti antene) zavržemo točke (m/2, n) = (9,1), (11,1), (12,1), (13,1), (13, 2), (14, 2), (16, 3), (17, 3), (19, 4) in (20, 4). Preostale se s teorijo ujemajo zelo dobro, s čimer smo tudi z numeričnim računom potrdili teorijo in to na 4 rede velikosti natančno.
vii Numerična študija negausovskih realno simetričnih matrik
Izhodišče za to študijo je simetrična matrika A = (A^) (i,j = 1 ali 2) iz enačbe (ii) s porazdelitveno funkcijo ga(a) diagonalnih ter gb(b) izvendiagonalnih matričnih elementov. Tej matriki smo za različne porazdelitvene funkcije izračunali porazdelitev po razmikih med sosednjima lastnima vrednostima P(S) in opazovali njeno obnašanje. Kot že omenjeno, je taka P (S) pri normaliziranih ga(a) in gb(b) avtomatično normalizirana, medtem ko njen prvi moment ni in ga dodatno normaliziramo
oo
Pn(S) = B P(BS), z B = / x P(x) dx. (xx)
o
Uvedli smo škatlasto (enakomerno)
gJa) = —, če \a\ < a0, sicer 0, (xxi)
y w 2a0
gb(b) = j, če |6| < bo, sicer 0, (xxii)
eksponentno
132
Daljši slovenski povzetek
10.000 F
1.000
n = 5
0.001 r<
10
n=4
20
30
40
50
60
kR
10.0000 1.0000
0.1000
0.0100
0.0010
0.0001
n=6
n=5
&=4
10
20
30
40
50
60
Slika xi: Koeficienti dinamičnega tuneliranja iz regularnega stanja (m/2,n) v kaotično morje za kvantna števila n = 1 do n = 5 (eskperiment) in n = 6 (numerika) versus kR za gobasti biljard z a/R = 10/19. Prikazana je teoretična napoved (’+’-i, povezani v krivuljo za fiksen n) iz enačbe (xviii) in eksperimentalni (zgornja) oziroma numerični rezultati (spodnja slika). Predstavljeni so na način, da so kvantna števila m/2 izpisana na sliki in povezana s pikčasto črto za fiksno kvantno število n. Pri eksperimentalnih in numeričnih rezultatih so vrisane tudi napake, ki pomenijo standardne napake povprečja. Črtkasta črta na vsaki sliki označuje eksperimentalno oziroma numerično resolucijo za 7.
vii. Numerična študija negausovskih realno simetričnih matrik
133
Cauchy-Lorentzovo
9a(a)
Wl + ^)
9b(b)
7T&0(1 + ^)
(xxiii)
(xxiv)
ter singularno krat ekponentno porazdelitev matričnih elementov
ga(a) = Ca\a\~^ae~XM, gb(b) = Cb\b\~ßbe~XbW, (xxv)
z normalizacijsko konstanto d = A-"w/(2r(l - ^)). Z numeričnih testom na 2 x 2 matrikah smo potrdili analitične rezultate (Grossmann and Robnik, 2007b).
Problem smo posplošili na večje matrike z dimenzijo N = 120. Hoteli smo testirati HW obnašanje (Hackenbroich and Weidenmüller, 1995), ki pravi, da pri (I) končnih in (II) omejenih porazdelitvenih funkcijah ga(a) in gb(b) vedno dobimo univerzalno GOE obnašanje, ne glede na podrobnosti porazdelitev. V primeru škatlaste in eksponentne porazdelitve smo potrdili napoved in za P(S) dobili Wignerjevo porazdelitev oziroma njen eksakten ekvivalent v oo dimenzijah. Pri škatlasti porazdelitvi matričnih elementov je prehod zelo hiter in se zgodi že do N = 7, medtem ko je pri eksponentni počasnejši, a je pri N = 120 že popolnoma končan. Zanimiva je Cauchy-Lorentznova porazdelitev (xxiv), ki nima prvega momenta in posledično njene lastne vrednosti niso omejene na končen interval (slika xii levo). Tu ne dobimo GOE obnašanja, kar vidimo na sliki xii desno. Če pa porazdelitvenim funkcijam v (xxiv) pri izbranih a in b odrežemo repe, se takoj pojavi GOE obnašanje. Oboje je v skladu z napovedjo (Hackenbroich and Weidenmüller, 1995).
Slika xii: Numeriˇcni rezultati za ansamble s Cauchy-Lorentzovo porazdelitvijo matriˇcnih elementov iz enaˇcbe (xxiv) z a0 = ?2/2 in b0 = 1/2. M je ˇstevilo matrik tega ansambla. Levo: gostota lastnih vrednosti za N = 120, M = 103, kjer je jasno vidno, da ni omejena na konˇcen interval. Desno: porazdelitev po sosednjih razgrnjenih lastnih vrednostih za N = 120, M = 103, kjer je razgrnitev narejena s fenomenoloˇskim pravilom unf = 20. Tanka krivulja je Wignerjeva porazdelitev za primerjavo.
Še bolj zanimivo pa je obnašanje v primeru singularne krat eksponentne porazdelitve matričnih elementov iz enačbe (xxv), ki že pri N = 2 za Aa = VŽ, Xb = 1, ßa = 0 in fib = 1/2 (slika xiii zgoraj levo) obljublja nov pojav, imenovan potenčno odbijanje sosednjih nivojev z eksponentom manjšim od 1 (ang. fractional power law level repulsion).
1
1
_
_
134
Daljši slovenski povzetek
Slika xiii: Numerični rezultati za ansamble s singularno krat eksponentno porazdelitvijo matričnih elementov iz enačbe (xxv) z Aa = y/2, Xb = 1, /x„ = 0 in fib = 1/2. M je število matrik tega ansambla. Levo zgoraj: N = 2 in M = 107, kjer je razgrnitev narejena s povprečjem po obeh lastnih vrednostih (vrisana je tudi teoretična vrednost (Grossmann and Robnik, 2007b)). Desno zgoraj: N = 3 in M = 105, kjer je razgrnitev narejena s povprečjem po vseh treh lastnih vrednostih. Sredina levo: N = 120 in M = 5 x 103, razgrnitev s fenomenološkim pravilom unf = 20. M je število matrik tega ansambla. Sredina desno: fib = 0.95 in M = 2 x 103, sicer isto kot na srednji levi sliki. Spodaj desno: ßb = 0.99 in M = 2 x 103, sicer isto kot na srednji levi sliki. Spodaj desno: fib = 0.9999 in M = 2 x 104, sicer isto kot na srednji levi sliki. Za lažjo primerjavo rišemo tudi Wignerjevo in na zadnji sliki Poissonovo porazdelitev.
Presenetljivo je, da pri istih parametrih že pri N = 3 opazimo prehod v linearno odbijanje nivojev in posledično v Wignerjevo porazdelitev pri N = 120, pri čemer je porazdelitev lastnih vrednosti polkrožna (ang. semicircle law), kar izpolnjuje pogoje HW napovedi.
Ko povečujemo ßb proti neintegrabilni singularnosti ßb = 1, se P(S) spreminja zelo močno in zdi se, da je spet prisotno potenčno odbijanje sosednjih nivojev z eksponentom manjšim od 1. Opazimo prehod P(S) od GOE pri majhnih vrednostih /ib k Poissonovi
vii. Numerična študija negausovskih realno simetričnih matrik___________135
(eksponenti) porazdelitvi, ko smo dovolj blizu ßb (slika xiii). V vmesnem območju pa se nakazuje željeno potenčno odbijanje.
136 Daljši slovenski povzetek
Izjava o avtorstvu
Izjavljam, da sem v predloˇzeni doktorski disertaciji uporabljal rezultate lastnega raziskovalnega dela.
Ljubljana, 29. september 2008.
Gregor Vidmar